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

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

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

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

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

We’ve compiled them here divided into four categories:

## Strategies for understanding a problem

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

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

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

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

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

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

## Identify important and extraneous information

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

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

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

## Schema approach

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

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

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

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

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

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

## Visualizing

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

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

## Guess and check

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

## Find a pattern

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

## Work backward

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

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

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

## Documenting working out

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

## Check along the way

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

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

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

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

Here are some checking strategies you can promote:

## Check with a partner

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

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

## Fixing mistakes

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

## Need more help developing problem solving skills?

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

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## 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 .

## Try Our Free Confidence Boosters

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.

## Get Khanmigo

The best way to learn and teach with AI is here. Ace the school year with our AI-powered guide, Khanmigo.

For learners     For teachers     For parents

## 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

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
• 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.

• “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

Teaching Math and More

A website that teaches math lessons and more.

## Strategies for Math Problem Solving

How do you teach strategies for solving math word problems?

Is there a step by step problem solving method that my students can use?

Do your students struggle to solve math word problems? Students often find it difficult to understand what to solve, how to start and find out the unknown. Solving math word problems doesn’t have to be hard. Teaching students how to solve math word problems is important. There are strategies for math problem solving that they can use today!

There are five strategies for math problem solving to word problems that you can teach your students in thirty minutes class. Before introducing these skills make sure you have reviewed how to read word problems first .  The second step in the problem solving process is to teach strategies that will help your students become better problem solvers. Try one or all of them today!

## 1. Drawing a Picture or Diagram.

This is a great strategy to use with visual learners. Students who are visual learners process information that they can see better than information that they hear. Drawing a picture helps them see the problem.

Here’s an example of using the strategy of a picture. What’s the problem tell us? There are four apple juice boxes in the cooler and those apple juice boxes are 1/3 of the juice boxes in the cooler. (Also Step 1) Draw the Problem. Draw 4 apple juice boxes. Say these are 1/3 of the juice boxes. Draw one circle around the 4 apple juice boxes, and then draw 2 empty circles. Question what would go in the other circles and how to get to the correct answer. This is great for math chats about the possibilities.

## 2. Find a Pattern.

Students should list the information already given in the problem. This list should reveal some very critical information about the problem. Examine the list of information for a pattern. What looks alike in the numbers? Does it repeat? Does it double? After finding the pattern, students should be able to identify the answer to the word problem.

## 3. Guess and Check.

The strategy is exactly like the name. Students guess the answer and then check their guess to fit the conditions of the problem. It’s a simple strategy, but very powerful to get students thinking.

## 4. Make a List.

This strategy is one of the most powerful ones. Students decide what information goes on the list from the word problem given. Organize the list by categories and make sure all the pieces of the problem are on the list. Lastly have students review the information that they organized on a list. Does it make sense? Can you reach a conclusion to solve the problem?

## 5. Use Reasoning.

To use reasoning students first need to organize the information given into a chart. Examine the relationships between the numbers. Think about the data and form a logical conclusion. Students may have to eliminate information to find the answer. Reasoning is not always easy to teach. Here are some questions to help guide students through using reasoning.

• Does the information make sense?
• What do these numbers have in common?
• Is there a pattern or relationship between the numbers?
• What can you conclude about the information?
• Does this word problem ask you to find something?

The most important thing you can do when teaching strategies for math problem solving is share as many as possible. You are teaching your students how to become problem solvers. The more strategies they know, the more independent and confident in problem solving they will become. As students become fluent problem solvers, they will be able to solve any word problem.

Try one or all the strategies and download the problem solving guide today!

## Hi I’m Kelly!

Hello! I'm Kelly McCown, the Teacher Author and Math Consultant behind this website. Thank you for taking the time to learn more about me and why I share classroom resources with fellow teachers. I started as a 5th grade teacher over 16 years ago. I loved getting to teach in a K-6 setting and be an advisor to the drama club. I moved to middle school and taught 6th, 7th, 8th grades and Algebra 1 Honors. I was a Middle School Math Club Coach for 3 years. I've had teaching certificates in elementary {K-6}, middle {English 5-9 and Mathematics 5-9}, and high school {Mathematics 6-12}. In 2013 I became a teacher author and started creating math curriculum for other teachers. I love teaching math to Elementary and Middle school students. Helping students conquer Math is something I take pride in. 100% of my Algebra I Honors students passed the state end of course … read more

## 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!

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!

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

## Getting the Most from Each of the Problem Solving Activities

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

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

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

## Shametria Routt Banks

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## 2 Responses

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

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

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Maneuvering the Middle

Student-Centered Math Lessons

## Math Problem Solving Strategies

How many times have you been teaching a concept that students are feeling confident in, only for them to completely shut down when faced with a word problem?  For me, the answer is too many to count.  Word problems require problem solving strategies. And more than anything, word problems require decoding, eliminating extra information, and opportunities for students to solve for something that the question is not asking for .  There are so many places for students to make errors! Let’s talk about some problem solving strategies that can help guide and encourage students!

## 1. C.U.B.E.S.

C.U.B.E.S stands for circle the important numbers, underline the question, box the words that are keywords, eliminate extra information, and solve by showing work.

• Why I like it: Gives students a very specific ‘what to do.’
• Why I don’t like it: With all of the annotating of the problem, I’m not sure that students are actually reading the problem.  None of the steps emphasize reading the problem but maybe that is a given.

## 2. R.U.N.S.

R.U.N.S. stands for read the problem, underline the question, name the problem type, and write a strategy sentence.

• Why I like it: Students are forced to think about what type of problem it is (factoring, division, etc) and then come up with a plan to solve it using a strategy sentence.  This is a great strategy to teach when you are tackling various types of problems.
• Why I don’t like it: Though I love the opportunity for students to write in math, writing a strategy statement for every problem can eat up a lot of time.

## 3. U.P.S. CHECK

U.P.S. Check stands for understand, plan, solve, and check.

• Why I like it: I love that there is a check step in this problem solving strategy.  Students having to defend the reasonableness of their answer is essential for students’ number sense.
• Why I don’t like it: It can be a little vague and doesn’t give concrete ‘what to dos.’ Checking that students completed the ‘understand’ step can be hard to see.

## 4. Maneuvering the Middle Strategy AKA K.N.O.W.S.

Here is the strategy that I adopted a few years ago.  It doesn’t have a name yet nor an acronym, (so can it even be considered a strategy…?)

UPDATE: IT DOES HAVE A NAME! Thanks to our lovely readers, Wendi and Natalie!

• Know: This will help students find the important information.
• Need to Know: This will force students to reread the question and write down what they are trying to solve for.
• Organize:   I think this would be a great place for teachers to emphasize drawing a model or picture.
• Work: Students show their calculations here.
• Solution: This is where students will ask themselves if the answer is reasonable and whether it answered the question.

## Ideas for Promoting Showing Your Work

• White boards are a helpful resource that make (extra) writing engaging!
• Celebrating when students show their work. Create a bulletin board that says ***I showed my work*** with student exemplars.
• Take a picture that shows your expectation for how work should look and post it on the board like Marissa did here.

## Show Work Digitally

Many teachers are facing how to have students show their work or their problem solving strategy when tasked with submitting work online. Platforms like Kami make this possible. Go Formative has a feature where students can use their mouse to “draw” their work.

If you want to spend your energy teaching student problem solving instead of writing and finding math problems, look no further than our All Access membership . Click the button to learn more.

Students who plan succeed at a higher rate than students who do not plan.   Do you have a go to problem solving strategy that you teach your students?

Editor’s Note: Maneuvering the Middle has been publishing blog posts for nearly 8 years! This post was originally published in September of 2017. It has been revamped for relevancy and accuracy.

## Problem Solving Posters (Represent It! Bulletin Board)

Check out these related products from my shop.

October 4, 2017 at 7:55 pm

As a reading specialist, I love your strategy. It’s flexible, “portable” for any problem, and DOES get kids to read and understand the problem by 1) summarizing what they know and 2) asking a question for what they don’t yet know — two key comprehension strategies! How about: “Make a Plan for the Problem”? That’s the core of your rationale for using it, and I bet you’re already saying this all the time in class. Kids will get it even more because it’s a statement, not an acronym to remember. This is coming to my reading class tomorrow with word problems — thank you!

October 4, 2017 at 8:59 pm

December 15, 2017 at 7:57 am

Hi! I am a middle school teacher in New York state and my district is “gung ho” on CUBES. I completely agree with you that kids are not really reading the problem when using CUBES and only circling and boxing stuff then “doing something” with it without regard for whether or not they are doing the right thing (just a shot in the dark!). I have adopted what I call a “no fear word problems” procedure because several of my students told me they are scared of word problems and I thought, “let’s take the scary out of it then by figuring out how to dissect it and attack it! Our class strategy is nearly identical to your strategy:

1. Pre-Read the problem (do so at your normal reading speed just so you basically know what it says) 2. Active Read: Make a short list of: DK (what I Definitely Know), TK (what I Think I Know and should do), and WK (what I Want to Know– what is the question?) 3. Draw and Solve 4. State the answer in a complete sentence.

This procedure keep kids for “surfacely” reading and just trying something that doesn’t make sense with the context and implications of the word problem. I adapted some of it from Harvey Silver strategies (from Strategic Teacher) and incorporated the “Read-Draw-Write” component of the Eureka Math program. One thing that Harvey Silver says is, “Unlike other problems in math, word problems combine quantitative problem solving with inferential reading, and this combination can bring out the impulsive side in students.” (The Strategic Teacher, page 90, Silver, et al.; 2007). I found that CUBES perpetuates the impulsive side of middle school students, especially when the math seems particularly difficult. Math word problems are packed full of words and every word means something to about the intent and the mathematics in the problem, especially in middle school and high school. Reading has to be done both at the literal and inferential levels to actually correctly determine what needs to be done and execute the proper mathematics. So far this method is going really well with my students and they are experiencing higher levels of confidence and greater success in solving.

October 5, 2017 at 6:27 am

Hi! Another teacher and I came up with a strategy we call RUBY a few years ago. We modeled this very closely after close reading strategies that are language arts department was using, but tailored it to math. R-Read the problem (I tell kids to do this without a pencil in hand otherwise they are tempted to start underlining and circling before they read) U-Underline key words and circle important numbers B-Box the questions (I always have student’s box their answer so we figured this was a way for them to relate the question and answer) Y-You ask yourself: Did you answer the question? Does your answer make sense (mathematically)

I have anchor charts that we have made for classrooms and interactive notebooks if you would like them let me me know….

October 5, 2017 at 9:46 am

Great idea! Thanks so much for sharing with our readers!

October 8, 2017 at 6:51 pm

LOVE this idea! Will definitely use it this year! Thank you!

December 18, 2019 at 7:48 am

I would love an anchor chart for RUBY

October 15, 2017 at 11:05 am

I will definitely use this concept in my Pre-Algebra classes this year; I especially like the graphic organizer to help students organize their thought process in solving the problems too.

April 20, 2018 at 7:36 am

I love the process you’ve come up with, and think it definitely balances the benefits of simplicity and thoroughness. At the risk of sounding nitpicky, I want to point out that the examples you provide are all ‘processes’ rather than strategies. For the most part, they are all based on the Polya’s, the Hungarian mathematician, 4-step approach to problem solving (Understand/Plan/Solve/Reflect). It’s a process because it defines the steps we take to approach any word problem without getting into the specific mathematical ‘strategy’ we will use to solve it. Step 2 of the process is where they choose the best strategy (guess and check, draw a picture, make a table, etc) for the given problem. We should start by teaching the strategies one at a time by choosing problems that fit that strategy. Eventually, once they have added multiple strategies to their toolkit, we can present them with problems and let them choose the right strategy.

June 22, 2018 at 12:19 pm

That’s brilliant! Thank you for sharing!

May 31, 2018 at 12:15 pm

Mrs. Brack is setting up her second Christmas tree. Her tree consists of 30% red and 70% gold ornaments. If there are 40 red ornaments, then how many ornaments are on the tree? What is the answer to this question?

June 22, 2018 at 10:46 am

Whoops! I guess the answer would not result in a whole number (133.333…) Thanks for catching that error.

July 28, 2018 at 6:53 pm

I used to teach elementary math and now I run my own learning center, and we teach a lot of middle school math. The strategy you outlined sounds a little like the strategy I use, called KFCS (like the fast-food restaurant). K stands for “What do I know,” F stands for “What do I need to Find,” C stands for “Come up with a plan” [which includes 2 parts: the operation (+, -, x, and /) and the problem-solving strategy], and lastly, the S stands for “solve the problem” (which includes all the work that is involved in solving the problem and the answer statement). I find the same struggles with being consistent with modeling clearly all of the parts of the strategy as well, but I’ve found that the more the student practices the strategy, the more intrinsic it becomes for them; of course, it takes a lot more for those students who struggle with understanding word problems. I did create a worksheet to make it easier for the students to follow the steps as well. If you’d like a copy, please let me know, and I will be glad to send it.

February 3, 2019 at 3:56 pm

This is a supportive and encouraging site. Several of the comments and post are spot on! Especially, the “What I like/don’t like” comparisons.

March 7, 2019 at 6:59 am

Have you named your unnamed strategy yet? I’ve been using this strategy for years. I think you should call it K.N.O.W.S. K – Know N – Need OW – (Organise) Plan and Work S – Solution

September 2, 2019 at 11:18 am

K now N eed to find out O rganize (a plan – may involve a picture, a graphic organizer…) W ork S ee if you’re right (does it make sense, is the math done correctly…)

I love the K & N steps…so much more tangible than just “Read” or even “Understand,” as I’ve been seeing is most common in the processes I’ve been researching. I like separating the “Work” and “See” steps. I feel like just “Solve” May lead to forgetting the checking step.

March 16, 2020 at 4:44 pm

I’m doing this one. Love it. Thank you!!

September 17, 2019 at 7:14 am

Hi, I wanted to tell you how amazing and kind you are to share with all of us. I especially like your word problem graphic organizer that you created yourself! I am adopting it this week. We have a meeting with all administrators to discuss algebra. I am going to share with all the people at the meeting.

I had filled out the paperwork for the number line. Is it supposed to go to my email address? Thank you again. I am going to read everything you ahve given to us. Have a wonderful Tuesday!

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## 9 Math Intervention Strategies for Struggling Students

Mathematics has potential to inspire wonder and foster creativity. It is far deeper than counting and arithmetic; math can provide irrefutable proof of grand ideas and predict new scientific discoveries.

As educators usher in a new generation of future problem solvers, a challenge to overcome today relates to current math proficiency levels . But fear not! No matter where students are on their math journeys, there are strategies educators can use to maximize and accelerate their growth.

## What Is Math Intervention?

In the broadest terms, math interventions are ways to help students who are behind in their math learning. However, for many educators, the meaning of math intervention is the support provided to students who are two or more grade levels behind in a math topic. Many schools offer dedicated classes—frequently with fewer students and more than one teacher—to support students who would benefit from intensive math intervention.

## Tier 2 Intervention and Tier 3 Intensive Intervention

The strategies described in this article can serve as Response to Intervention (RTI) math strategies. However, they do not differentiate among the needs for a multi-tiered system of supports for interventions within the core classroom (Tier 1), as supplemental intervention through differentiated instruction (Tier 2), or as intensive intervention that can occur in either the core classroom or in a traditional pull-out classroom (Tier 3).

• Tier 1: Students receive research-based instruction within their core classroom.
• Tier 2: Students receive targeted supplemental intervention using differentiated instruction.
• Tier 3: Students receive individualized, intensive intervention within their core classroom or in a pull-out model.

All students have different learning needs. What works for one student may not work for another, and what works for a particular math skill or problem may not work for another. Moreover, the supports that your students need may look different depending on classroom capacity or the organization of specialized intervention support within your school or district. Feel free to try the ideas below, adapting them to your individual students.

## Planning Lessons for Students Who Are Struggling

Explore this list of math intervention strategies before giving a lesson, as they can be used in whole-class or small-group math instruction and guide what problems to use in the first place.

## Strategy 1: Account for Student Strengths

So your student doesn’t like math? Or maybe doesn’t seem to make any progress with the topic you’re teaching? Well, what do they like? Account for the whole student. Ask about their family, hobbies, and entertainment. Maybe there’s a class they particularly love or even a previous math topic that interested them. Consider creating an interest survey where students share their feelings about math, which can extend to what interests them outside of math, too. Look for ways to connect what your students love to the math topic that’s confounding them.

This can also be an opportunity to introduce students to new ideas and subjects that they didn't even know would be of interest to them. If your math program includes videos or simulations , encourage your students to explore. If it doesn't, there are still plenty of free options online. Listen to your students and figure out what personally engages them so you can tap into what they care about.

This isn't a simple task for you, the teacher! You may need to research details about your students’ interests and seek out the mathematical connections. It may help to browse our full library of free resources , including math activities that relate to fashion , sports , business , and art , to name a few.

What works for one student may not work for another, and what works for one math topic may not work for another.

## Strategy 2: Use Schema-Based Instruction

Word problems can leave students across all grade levels wondering where to even start. The combination of parsing English and manipulating mathematical concepts can be daunting, especially for multilingual learners . One evidence-based strategy is to create a schema, or an underlying blueprint or structure that you can introduce students to and continually revisit when working through word problems. For example, consider the following word problems:

• Alice has 81 fish that she has to place evenly into 9 buckets. How many fish are in each bucket?
• Rosario wants to hang 25 artworks into 5 galleries so that every gallery has the same number of artworks. How many artworks go in each gallery?
• A pet shelter has 18 cat treats that it wants to give out evenly to the 6 cats in the shelter. How many treats will each cat get?

All of these problems can be described using a schema that describes the general concept of division:

• Total ÷ Groups = Number per group

Once you have a schema, you can refer back to it to help students illustrate problems and adapt new word problems to the same schema. It may help to draw the schemas , create posters for the ones that appear most frequently, or—if your students are ready—see if they can define their own schemas.

## Strategy 3: Peer Tutoring

Want to know one of the best ways to learn math? Teach it. Seriously; ask math teachers of all stripes, and they can attest that having to explain an idea to others makes them learn it more soundly than they ever thought possible. This can be a scary strategy to employ when the students don't seem to have mastered a concept yet themselves, but plenty of learning can happen through the teaching process. Having students talk to each other about math fosters rich math conversations. It also helps students identify classmates who are strong counterparts in terms of helping them when they're stuck.

Pair students up and have them teach a concept to each other. Be sure to give them tools to tutor another student even if they’re struggling themselves. For example, you could provide (or have students create!) cards with numbers, equations, images, or vocabulary terms. The “tutor” can select the cards, and the “tutee” can sort them and form rules that describe all of the cards in each group. After 5–10 minutes, have students reverse roles: the tutor becomes the tutee. Not only does this strategy improve math understanding for both students; it bolsters confidence and attitudes about math, to boot!

If your students could use some practice with division, try out the card sort activity below, which comes from Math 180 , our math intervention solution for students in Grades 5 and up.

## Strategy 4: Practice Fact Retrieval

Part of what stops a lot of students from progressing in math is frustration over not knowing and constantly getting stuck on math facts. After all, if you can’t quickly add and multiply, how can you be expected to solve larger problems that rely on those facts?

Modern evidence suggests that math facts should not be the focus of lessons and are improved through general practice. However, some evidence shows that setting aside some time for practicing math facts—around 10 minutes per day—can reap large dividends in terms of confidence in math class and growing in math competency beyond fact fluency. There are endless ways to practice math facts with students, including video games , board and card games, and the strategy that follows: Cover-Copy-Compare.

Fact fluency can also be thought about in the context of problem solving. When solving math problems, don't quickly correct an arithmetic error and move on. Instead, spend time on it and have students discuss their thought processes. Consider employing Number Talks, five- to fifteen-minute conversations around computation problems that are solved mentally, which are great at getting students to work naturally on their fact fluency and speed of recall.

## Strategy 5: Cover-Copy-Compare

This is an intervention that offers a specific evidence-based activity to practice fact retrieval . In preparation, create a worksheet with around 10 math facts. Have the list of sums or products (for example) lined up along one side of the paper. Let students study the facts and, when they’re ready, cover the lined-up numbers and try to recreate the list. When they’re done, compare the list they generated to the original list.

Mark the facts correct or incorrect. If any are incorrect, repeat the procedure of covering, copying, and comparing until all the facts are solved correctly. This task helps create a definable and manageable goal for students—get all of the facts correct—no matter their current math abilities. You can differentiate the task by having students answer orally instead, which has the added benefit of being quicker. This routine improves math accuracy both across general education and special education , and can apply to math ideas beyond just math facts! For example, this strategy could be used to build fluency with greater than or less than symbols.

## Strategies to Use While Teaching

Try this list of math interventions while giving a lesson, as they can be tools to help individual students—or even full classes!—who seem to be stuck on a problem.

## Strategy 6: Employ Metacognitive Strategies

Research has shown time and time again that if you can get students to think critically about their own mathematical thinking, there is an opportunity to grow. On the surface, the problem may seem to be an understanding of math concepts, but the deeper problem may in fact be the students’ mindsets.

Encourage students to tell their own math stories. What was math like in their household and in previous classes? Intentionally draw your students’ attentions not just to what the math concepts are, but how they feel about them. Have students pause to reflect on how they feel and what thoughts are crossing their minds. Are they thinking thoughts like “I’ll never get this” or “There’s nothing I can do about it?” Present ways to combat this thinking, such as taking a break, asking for help, or brainstorming a new strategy to try.

## Strategy 7: Verbalize Thought Processes

Research shows that the most successful math interventions are explicit and systematic. One way to do that is to verbalize thought processes. In other words, as students think about how to solve a problem, probe them to say what they are thinking aloud. It is important here to listen patiently and without judgment even if their mathematical language is inexact or their reasoning is imperfect. When you hear the full process, it can help you identify specifically how to intervene. Perhaps they understand a larger idea but get stuck on the arithmetic. Or perhaps they understand what a problem is asking but shut down as soon as they encounter a fraction.

It can work the other way, too. As you solve a problem, verbalize your thought process. Let students see the steps you take to work through a problem and model precise mathematical language and reasoning.

## Strategy 8: Fast Draw

Fast Draw is a learning strategy devised by Cecil D. Mercer and Susan P. Miller around 30 years ago to help students with learning disabilities in solving math word problems. The letters of Fast Draw are a mnemonic for the steps:

• Find what you are solving for: Look for the question mark and underline what you are trying to solve for.
• Ask yourself what information is given: Read the whole problem and look for what information is already provided.
• Set up the equation: Write the equation with numbers and symbols in the correct order.
• Tie down the equation: Say out loud what the operation is and what it means. If you can, solve the problem. It may help to draw pictures.
• Discover the sign. Find the sign and say it out loud.
• Read the problem. Say the problem out loud.
• Answer the problem, or draw.
• Write the answer to the problem.

Here’s a resource from James Madison University that provides details on each step, along with examples. Students struggling in mathematics often become passive when faced with word problems , and Fast Draw offers a concrete strategy that can help them become active and work through word problems on their own. The strategy not only helps bolster math achievement, but math attitudes, too, especially for students with learning disabilities.

## Strategy 9: Use Multiple Representations

Multiple representations are important well beyond math intervention. They help students perceive math concepts in different ways and form important generalizations. However, they can serve a more targeted purpose for a student who needs targeted help. Showing math representations in different ways gives students a variety of mental models to consider, making comprehension more likely. In other words, it helps students “see” the math, even when one representation confuses them. The card sort activity from Strategy 3: Peer Tutoring is one way you can prepare specific multiple representations for students to compare.

Even concepts that seem simple to you can be complex to your students. Since everybody without visual impairments is a visual learner , lean into multiple visual representations to demystify ideas. But be intentional in which representations you show. If the representations are hard to interpret or come across as disconnected, you risk confusing students more !

Looking to unlock mathematical learning in the students who need it most? Explore our math intervention programs for students in Grades 5–12.

## Using Math 180 to Accelerate Math Learning for All

Math 180 is a math intervention program for students in grades 5 and up. Find out how to use the math program to accelerate math learning for all students.

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January 12, 2022

## 7 Math Small-Group Instructional Strategies

These small group instructional strategies for the math classroom will help ensure your small groups in math class are effective for you and beneficial for students.

Richard Blankman

December 10, 2021

## Do Your Students Have a Fixed Mindset or Growth Mindset? The Answer Is Both

Get the truth about fixed vs. growth mindsets and try our strategies to foster an environment where students feel empowered to learn.

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## Writing and choosing problems for a popular high school mathematics competition

• Original Paper
• Open access
• Published: 29 March 2022
• volume  54 ,  pages 971–982 ( 2022 )

• Robert Geretschläger 1 &
• Lukas Donner   ORCID: orcid.org/0000-0003-1263-1478 2

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In this paper, we consider the issues involved in creating appropriate problems for a popular mathematics competition, and how such problems differ from problems typically encountered in a classroom. We discuss the differences and similarities in school curricula versus the generally agreed upon topics encountered in international competitions. The question of inspiration for the development of competition problems is dealt with from the standpoint of the problem author, while aspects related to the motivation of the contest participant, objective and subjective problem difficulty and mathematical precision in mathematics competitions are also discussed.

Avoid common mistakes on your manuscript.

## 1 Introduction

The problems typically posed in popular or inclusive mathematics competitions are similar to, but not identical to, problems regularly considered in a typical mathematics classroom. The argument can be made that competition-style problems should be used in the classroom setting a bit more than is currently the case (see, for instance, Geretschläger, 2017 , 2020 ; Applebaum & Gofman, 2015 ), but there are certainly marked differences. These result quite naturally from their differing goals. While many researchers are dealing with the goals of textbook problems from several perspectives, the situation is not as yet that well discussed for competition-style problems. Kontorovich ( 2015 ) asked several adults from the Competition Movement Footnote 1 and reported that this group formulated four goals that can be achieved by competition problems. Besides learning meaningful mathematics, further pedagogical goals are the strengthening of students’ positive attitudes towards mathematics, the creation of cognitive challenges, and adding an element of surprise for students. Unlike a typical textbook problem, the intent behind a competition problem is neither to teach a foundational concept nor to check whether the concept has been understood or to solidify such comprehension. While these aspects may also apply to its consideration, the main reason for a competition problem to exist is motivational. When we are posing a problem for a competition with a large number of non-specialist participants, the most important aspect is always consideration of the question, Is this interesting? Will the problem, as set here, capture the imaginations of the students once they are exposed to it?

Of course, there is an underlying assumption that students willingly engage with all problems that come their way in school, but this is unfortunately not always the case. There are things that need to be understood and practised on a basic level before it is possible to move on to the applications that most people would consider ‘interesting’. On the other hand, it should be possible to assume that each problem posed in a competition is worthy of being thought about in its own right, simply because its solution is rewarding in its satisfaction of intellectual curiosity. A good competition question will always find a large number of participants invested enough to want to engage in finding its solution, simply because the problem is intrinsically of interest to them. Reznick ( 1994 ) called this phenomenon inevitability: once you see the problem, you feel you must solve it.

The aim of this paper is to address the question of how to best create, refine and select such problems for inclusion in competitions. Currently, problem selection for competitions is not done with didactic research results in mind, and we strive to bring this element into play by emphasizing connections to current topics in education research, specifically with regard to word problem solving and problem posing. Furthermore, we summarize the state of the art in competition mathematics in order to offer the option to both the Competition Movement and the mathematics education community to build on this state for future research.

A specific challenge arises in selecting problems for a popular competition, where participants cannot be expected to be as self-motivated as for top-level competitions such as the Mathematical Olympiads (MO). The problems should be such that any reasonably receptive participant will want to engage with them, in the hope of experiencing the feeling resulting from a successful solution and therefore enriching their learning (Kenderov, 2006 ).

In the following, the matter of creating suitable problems will be considered through the lens of the International Mathematical Kangaroo (MK). One of us (Geretschläger) has been actively involved in the problem-selection process since 1998, and as group chair of the Student level (grades 11 and 12) for 20 of those years. Specifically, the discussions delineating the differences between potential problems to be chosen for this competition, from ‘textbook’ problems on the one extreme and ‘Olympiad’ problems on the other, have led to much fruitful reflection on this topic over the years. The research process for this paper was as follows: in a first phase, both authors, as actors and proven experts in the field, reflected on what they considered to be the most essential aspects in the creation and selection of tasks for the MK and agreed on a common list of factors, including curriculum, range of difficulty, precision, and packaging. In a second step, these aspects were reflected on in detail and linked to the current didactic discourse. The results of this two-step process are presented in Sects.  3 to 8 .

The MK is an international competition that has been organised since 1991. Recently, it has attracted more than six million participants annually. It is organised in more than 70 countries, with this number growing consistently.

The competition is organised in the following six levels: Pre-Écolier for grades 1 and 2, Écolier for grades 3 and 4, Benjamin for grades 5 and 6, Cadet for grades 7 and 8, Junior for grades 9 and 10, and Student for grades 11 and 12 (and 13 where applicable). The number of items in each paper varies by level from 15 to 30, and the time available to solve them also varies by level from 60 to 75 min. Each level has an equal number of multiple-choice items (1 out of 5) worth 3, 4 or 5 points, with the relative difficulty of the items rising with the number of points available. One quarter of the available points are deducted for incorrect answers, with no penalties imposed for omissions.

The problems are created and selected by an international group of experts at an annual meeting, at which all participating countries are represented. They are specifically designed to provide an interesting challenge for all levels of knowledge and talent.

This paper is organized as follows.

In Sect.  2 , we explicitly identify key differences between a competition problem and a textbook-problem. In this context, we compare problems of the two types with similar mathematical content. We also address the fact that the mathematical knowledge required to solve an item from the MK is often encountered in class at a younger age than that of the participants in the competition. In Sect.  3 , the focus is on competition curriculum. The typical restrictions in the competition curriculum, as well as the possibly missing connection to mathematics taught in the classroom, are often cited as negative elements by members of the mathematics education community (Kenderov, 2006 ), and we address the resulting tension. Also, we discuss the general prohibition of electronic tools in competitions. This goes against the current vogue in general education in many countries, whereby practical calculation in regular mathematics classwork is usually farmed out to a calculating device, resulting in yet another nuance of difference in the mathematical content of competitions and the classroom standard. The main focus of Sects.  4 – 6 is the creation of appropriate, interesting and not too difficult high-quality problems and distractors for the MK. Simultaneously, we embed the practitioner’s considerations in the theory of problem posing. We discuss the question of where the inspiration for an interesting popular competition problem comes from. Also, we discuss the difficulty of finding good, easy and interesting problems for competitions, and deal with the fact that it is not as difficult to find ‘hard’ problems that are also interesting. The qualities required for a problem to be considered interesting are precisely what tend to make them hard. This is true for solutions requiring multiple steps, a combination of ideas from different topic areas, or open-ended options for solutions. In considering problems at this end of the spectrum, there are therefore other matters that come into play. This leads to the matter of when a ‘hard’ problem is too hard for a popular competition, and when it is still appropriate. More specifically, we consider the line separating appropriate problems for the MK from those of advanced competitions that require special preparation, like the MO. In addition to the consideration of the maximum and minimum levels of difficulty and complexity of an appropriate MK problem, there is also the matter of the structure of the problems. When creating problems for mathematical competitions, the problem poser must be aware of the fact that “such a problem should be well-formulated and have no unnecessary givens; its formulation is supposed to be short and include some elements of innovation for a particular category of solvers” (Sharygin, 1991 , cited by Kontorovich & Koichu, 2016 ). In Sects.  7 and 8 we address these aspects—specifically the aspects of the appropriate brevity of problem formulations (i.e., precision) and the elements of innovation of a problem (i.e., abstractness or ‘packaging’). We also establish connections to research findings on word-problem solving. This aspect reminds us that the MK contains a very special subset of these problems. Some problem posers argue that the packaging of a problem is of great importance for improving the attractiveness of a task in order to engage as many students as possible, and we address the relevance of such arguments in Sect.  8 .

## 2 The essential difference between a textbook problem and a closely related competition problem

The following problems illustrate the difference between a good competition problem and an exercise in the classroom. Item A is taken from the Student paper of the MK 2018, where it was posed as a problem of medium difficulty. Item B is a problem for grade 9, taken from the textbook by Geretschläger et al. ( 2004 ). Both are problems from the same topic area, namely, triangles in the coordinate plane, and both deal with the mid-points of the sides of a triangle.

(A) The vertices of a triangle have the coordinates A(p,q), B(r,s) and C(t,u) as shown. The mid-points of the sides of the triangle are the points M(− 2,1), N(2,− 1) and P(3,2).

Determine the value of the expression p  +  q  +  r  +  s  +  t  +  u .

(B) We are given a triangle ABC with vertices A(1,− 3), B(3,4), C(− 1,2). Calculate the common point of the three medians of the triangle.

Depending on the level of sophistication of a student solving the textbook problem B, the expected solution will be attained by calculating the averages of the x- and y-coordinates of the three points or as the common point of two medians. Either way, there is standard knowledge that students are expected to apply as studied in the classroom and explained by a teacher or in a textbook.

For the competition problem, the mid-points of triangle sides also play a role, but in an unexpected way. It cannot be assumed that students have been exposed to such a situation before, and yet it is not so hard for them to find connections to facts learned in the classroom, and apply them in an original way. If they notice that the coordinates of any of the given points M, N and P can be expressed as the means of the unknown coordinates of the triangle vertices A, B and C, they obtain

which yields D as the correct answer. Nothing here could be considered mathematically difficult or sophisticated in any way. Nevertheless, the fact that the structure of the problem is of an unexpected nature makes the question a puzzle. Solving the puzzle requires an original idea. In this case, this is simply the collection of sums of the various unknowns in such a non-standard way that they can be replaced by known values. A student who finds this method, or some other original way to tackle the problem, has discovered something new for him- or herself.

This comparison gives an example of what is perhaps the defining difference between a competition problem and a classroom exercise. Mathematical tasks in classrooms aim at supporting students to engage in a range of mathematical activities with specific didactical goals, such as exploration, concept formation, or practising skills (Barzel et al., 2013 ). Many tasks are meant to be solved by applying learned facts in a standard way in a context of a type that has been previously encountered in order to practice. The competition problem, on the other hand, while also using the same basic tools, is meant to be solved by coming up with some (small) amount of original thought.

Certainly item A could conceivably be used in a classroom, but item B would typically not appear in any competition, as it does not offer a student outfitted with the requisite basic knowledge the opportunity to discover anything original.

Item A was presented for grades 11 and 12, whereas its content is typically taught in grades 9 or 10. A strong argument in favor of keeping the problems ‘age appropriate’ lies in the idea of making use of the tension between the recreational aspect of an interesting problem and its educational aspects. If students engage with a problem because they find it intrinsically interesting, there is a better chance that the learning process will yield the desired results. This could be interpreted to imply that it would be preferable to stick to subject matter close to the curricular material students are studying at any given time in proposing competition problems, but this severely limits the options for competitions. It can reasonably be expected that students will find something interesting to engage with if the solution of a problem requires material that they are well familiar with, and not just material they are currently studying. It is a constant struggle to find an appropriate balance between these two viewpoints in choosing competition tasks.

## 3 The matter of curriculum

Topics chosen for competitions tend to be taken from limited areas of mathematics. Also, the type of reasoning and calculation expected in solving such problems tends to be similarly limited. Sharich ( 2017 ) pointed out that these phenomena imply a crisis of substance, i.e., “the lack of challenging new mathematics in problems offered at Olympiads” (p. 72). However, there is a wide international consensus for these limitations that has developed over the last few decades, resulting in something frequently referred to as “modern elementary mathematics”. It can be defined as the union of certain elementary methods and the problems that can be solved by means of these methods (see Koichu & Andžāns, 2009 ). Kenderov ( 2006 ) emphasised that especially through the Competition Movement, this important part of our mathematical heritage is kept alive and further developed. It is interesting and informative to take a look at the reasons why modern elementary mathematics is so widely accepted in the competition community.

First, there is the matter of diverging national school curricula. Reduction to the intersection of the various national curricula eliminates such topics as calculus, complex numbers, set theory or statistical analysis for international competitions, to name just a few. Unfortunately, finding this intersection is not as obvious as might be suspected, and students with a background in schools that teach a curriculum close to the agreed upon ‘competition curriculum’ are at a competitive advantage in the world of the MO. Andreescu et al. ( 2008 ) pointed out that what most of the countries that are successful at international competitions “have in common are rigorous national mathematics curricula along with cultures and educational systems that value, encourage, and support students who excel in mathematics” (p. 1251).

Next, there is the matter of technology. In contrast to the decades-long trend of including digital technology in curricula all over the world (e.g., Sinclair et al., 2009 ), there is a wide consensus in the Competition Movement that technology should not be permitted in most competition settings. The main reason for this is the fact that number structure and number properties (prime numbers, perfect squares or higher powers of integers, unusual types of fractions, etc.) are considered to be a fertile ground from which interesting logical puzzles can be harvested. Shi ( 2012 ) described how a classical problem on certain digits of higher powers of integers can be solved easily with the use of digital technology. Technology has the potential to open up new routes for students to construct and comprehend mathematical knowledge and new approaches to problem-solving (Bray & Tangney, 2017 ). However, by permitting the use of technology at competitions, the focus of attention would be shifted from mathematical reasoning towards the “mathematical digital competency” defined by Geraniou and Jankvist ( 2019 ). There is a general feeling that this characteristic type of subtle thinking and reasoning in number theory is eliminated by hitting every number-based problem with the sledgehammer of technology.

As a third point, emphasis is given in competitions to thinking as opposed to calculation or direct application of learned facts. Clear preference is given to problems whose solutions require original insight on the part of the participant, while problems that rely heavily on facts and methods that can be studied in advance are pushed into the background. Methods relevant to the solution of problems in modern elementary mathematics are either not specific to any particular branch of mathematics (e.g., symmetry or equivalence), involve the analysis of very specific singular cases, or are very general methods from specific branches, such as the invariant method in combinatorics (see Engel, 1998 for a comprehensive collection of methods). One consequence of this feature is the fact that problems posed at competitions tend to be closer to puzzles in their structure than to applications to real-world situations.

The International Mathematical Olympiad (IMO) is often used as a benchmark for comparisons between competitions. This makes a lot of sense, since the IMO has the longest tradition of the high-level international competitions (since 1959), the largest number of participating countries (over 110 in recent years) and a wide level of acceptance. The problems developed for this specific competition are traditionally grouped into four categories, namely, Algebra, Combinatorics, Geometry and Number Theory (e.g. Djukić et al., 2011 ). While their meanings are subtly different from the standard usage in research mathematics, the consensus concerning these topic groups is quite clear.

Variations of these four IMO categories are often used to sort the problems posed in other competitions. At the MK, for instance, the categories used in problem selection are Algebra, Logic, Geometry and Number. While some of the most interesting problems do not slot readily into any of these categories, an attempt is usually made to have all four groups represented as equally as possible. This is, of course, due to the format of the competition, where many multiple-choice questions are to be answered under restrictive time constraints. The flavor of the questions should therefore be as wide-ranging as possible, while also being accessible to as many participants as possible (Akveld et al., 2020 ).

This criterion means that problem-creation is restricted a bit. Of course, problems that do not require any knowledge at all, and can be solved with original clever ideas, are particularly esteemed.

## 4 Creating a competition problem

Empirical research on problem posing is comprised of three main perspectives, namely, focusing on problem posing as a cognitive activity, as a learning goal or as an instructional approach (Liljedahl & Cai, 2021 ). Although the term problem posing is used inconsistently in mathematics education research when focusing on situations within mathematics lessons (Baumanns & Rott, 2021 ), the term is less ambiguous when addressing the creation of competition problems. How one develops the capacity to pose good problems is the central question in research on problem posing as a goal of mathematics instruction (Cai & Leikin, 2020 ). Within a survey study, Lee ( 2020 ) found that mathematical experts are rarely involved in studies on problem posing. As Kontorovich and Koichu ( 2016 ) pointed out, there currently exist only a handful of papers devoted to the principles of experts’ problem posing for mathematics competitions. Research in this area seems to have started with two Russian papers in the 1990s by Sharygin ( 1991 ) and Konstantinov ( 1997 ). According to Sharygin ( 1991 ), there are six general techniques used for creating new problems, which he names Reformulating, Chaining, Considering special cases, Generalizing, Varying the givens and Inquiry . As Kontorovich and Koichu ( 2016 ) pointed out, however, being aware of these techniques and carrying them out is not enough for posing a problem of high quality, as other studies with students and mathematics teachers as problem posers show. One reason may be the pool of familiar problems that are an important component of an expert’s knowledge base (Sharygin, 1991 ), a fact that can be implicitly found in problem posers’ papers, such as those by Klamkin ( 1994 ) or Reznick ( 1994 ). An individual new to the process of problem posing may well be fully cognizant of the useful methods but may not have the breadth of familiar problems at their disposal to refer to. Elgrably and Leikin ( 2021 ) underpinned this assumption via use of Problem-Posing-through-Investigation (PPI) tasks, a mathematical activity that combines problem posing and problem solving. The authors compared the performance in PPI (in a dynamic geometry environment) of members (or candidates) of the Israeli IMO team and mathematics majors who excelled in university mathematics. One finding was that the former “were more fluent, flexible and original and produced more complex problems with more complex auxiliary constructions” although the latter “took a geometry course with a specific focus on PPI” (Elgrably & Leikin, 2021 , p. 902). Recent research on mathematics competition problem posing focuses on the process of creating new Olympiad-style problems via empirical case studies of experts in problem posing and not just on self-reflection (e.g., Kontorovich & Koichu, 2016 ; Poulos, 2017 ). A main point made by Kontorovich and Koichu ( 2016 ) is that the solution of a problem must surprise even experts in order for them to be satisfied with the problem. These findings highlight the role of affect in experts’ mathematical problem posing, namely surprise, which is also a main characteristic of a good problem (Reznick, 1994 ). This aspect cannot be achieved by looking for problems in the expert’s pool of familiar problems if the problems in this pool are simply classified by the similarity of their solving strategies. An expert therefore requires (and uses) further principles of grouping problems inside their own personal pool.

Affect in experts’ mathematical problem posing was empirically treated by Kontorovich ( 2020 ), who interviewed problem posers from all types of mathematical competitions. He determined the existence of three problem-posing triggers , the outcomes of which are often the creation of problems that go on to be chosen for competitions. Two of these are emotionally inspired. These are instances in which the problem poser extracts mathematical phenomena either from activities derived from some aspect of modern elementary mathematics (e.g., problems, concepts, theorems, problem-solving methods) or from everyday tasks in which mathematical content was somehow beneficial. A third—extrinsic—trigger is mentioned by the experts as well. This results when they are asked to pose problems ‘here and now’, i.e., problems with partially predetermined properties that are required for a specific competition. The author points out that “there was a consensus among the participants about how difficult it is to pose in such situations and about the low quality of the resulting problems” (Kontorovich, 2020 , p. 402) in the latter case.

It should be pointed out at this juncture that there is a lack of research with respect to the creation of problems for the MK. For this type of competition, the design of the task is as important as the mathematical idea behind the problem. A main focus lies on the distractors (the ‘multiple choices’) of the problems. Often these distractors are meant to be used by the student to solve the problems, while incorrect answer options are sometimes meant to set a trap for a careless solver. One of us analysed tasks of the MK of 2018 with respect to the distractors in Andritsch et al. ( 2020 ) Footnote 2 . There, it is pointed out that problem posers should be consciously aware of the fact that some problems of the MK can simply be solved by using the convergence strategy according to Smith ( 1982 ) or applying particular test-wiseness strategies as defined by Millman et al. ( 1965 ). These strategies give advantages in solving multiple-choice problems without actually applying thematic knowledge. By trying to offer particularly ‘beautiful’ and ‘suitable’ incorrect answer options, the problem poser gives students the option of bypassing the solving process in obtaining the correct answer, and although the design of the competition—lots of multiple-choice problems and not a lot of time to reflect on them—encourages students to guess when the odds seem advantageous, most problem posers would agree that this choice should always be based on content related ideas. The actual use of test-wiseness strategies by successful participants at the MK was studied by asking Austrian winners of the 2018 competition (Donner et al., 2021 ).

## 5 What makes interesting easy problems so hard to find?

It is quite a multi-faceted challenge to create a problem that is simultaneously interesting and easy. This is the case for competitions in general, but it is especially true for the MK.

One reason for this difficulty may be intrinsic to one of the points raised by Kontorovich and Koichu ( 2016 ). If the level of innovation of a created problem is too small, the poser is not surprised by the content of the problem and thus does not classify it as an interesting problem. There may therefore be a lack of motivation on the part of a potential poser to create an easy problem from such a starting point. Furthermore, the poser can only rely on a limited part of his pool of familiar problems for this purpose. The poser may nevertheless feel obligated to create an easy problem ‘here and now’ due to time constraints, or due to being asked to do so by his or her peers.

What exactly is meant by an ‘easy’ problem in the MK?

There is wide-spread agreement that the 3-point problems should be easily solvable by all participants in the MK (Akveld et al., 2020 ). These problems should be accessible to students not generally interested in mathematics. The whole idea of the contest is popularisation, and all students should therefore have a real chance of achieving a feeling of success after solving and understanding something.

Regarding complexity, there is a broad consensus in the MK, that 3-point problems should not require more than one idea to solve and should ideally require only a single step of calculation or logical reasoning. These aspects are illustrated by the solution of the following 3-point item from the Student paper 2020.

(C) The sum of five three-digit numbers is 2664, as shown on the board. What is the value of $$A + B + C + D + E$$ ?

This task can be solved by the fundamental idea of reformulation . Footnote 3 A three-digit number $$ABC$$ can be written as $$100 \cdot A + 10 \cdot B + C$$ . Applying the observation that each of the five digits occurs as a unit- tens- and hundreds-digit exactly once, the given equation can be reformulated and the solution found by one step of calculation: $$111 \cdot ( {A + B + C + D + E} ) = 2664 {\text{ or }} A + B + C + D + E = 24$$ .

An alternative is to note that the sum of all five digits ends on the digit 4, as seen in the unit-digit of the sum. This means that there is a carry-over of 2 in order for the tens-digit to result from the same sum, and the sum of the digits must therefore be 24.

Without the application of one of these fundamental ideas, a student will not be able to solve the problem easily. (Of course, another way to solve this problem is pure guessing of a possible assignment of the variables, but this approach will take much more time in general.)

Solving this prototype of an easy problem does indeed involve one single idea and one step of calculation. Nevertheless, it is quite far from a typical textbook problem.

The problem can be described as interesting, because although it is not possible to determine any of the indeterminants A to E , their sum can be uniquely determined. (This basic idea occurs quite often in different problems spread out over all age categories and competition years.)

A second consensus broadly agreed upon within the MK community concerns the design of the 3-point problems. It is often the case in problem development for multiple-choice competitions that problems are designed in such a way that certain errors in reasoning or illegitimate simplifications will lead to one of the incorrect answer options. Such an offered solution is a trap for anyone following a specific incorrect (but anticipated) train of thought. This may very well be the intent of the problem poser, meant to make a problem interesting that seems to be a routine task at first glance. Such tactics are intentionally avoided in the 3-point section of the MK. Similarly, problems that involve distractors which ‘widen the spectrum’ of possible answers, like distractors of the form ‘none of these’ or ‘more than x’ , to questions about the number of possibilities for something, are avoided there. In general, such distractors can enrich the value of a problem and make it more interesting, because methods like ‘trying out’ distractors or finding x possibilities for something cannot ensure finding the correct solution to the problem. This reduces the number of tactical options a potential solver has at their disposal in dealing with a problem. It follows that these kinds of distractors are not suitable for problems meant to be easy.

Finally, as students are generally unfamiliar with problems involving aspects of more than one content category, straddling the borders between categories automatically increases the level of difficulty. Therefore, in order to be ‘easy’, a problem should not involve more than one subject. Quite often, however, it is exactly such interaction between content categories that makes a problem interesting. This means that problems of this kind—such as example A in Sect.  2 , which involves aspects of geometry and algebra—may not be difficult from a mathematical perspective, but the unfamiliarity of their content makes them difficult in the context of a popular competition. Problems of this type are perfectly well suited to the MK, but are not categorised as 3-point (i.e., easy) problems.

Putting together all these restrictions, it is perhaps less surprising that it is so hard to find competition problems that are both easy and interesting. There is a very fine line separating routine (textbook) problems and interesting but multiple idea/multiple step problems. By avoiding traps and certain all-too-obvious distractors and ideally sticking to a single content category, further restrictions have to be fulfilled. Considering all of this, there appears to be an inherent contradiction between the requirement that a problem should be ‘interesting’ and the requirement that it should be ‘easy’.

Despite the severely limited pool of problems fitting this description, problem posers are constantly challenged to find appropriate easy and interesting problems. As the annual problem sets confirm, they are often quite successful at overcoming the difficulties.

## 6 Hard problems in popular competitions

The most obvious reason for a problem to be rated as too hard for the MK involves the level of mathematics required for a solution. If a problem can be solved only by applying some higher-level results that may not be known to all participants, the problem will not be able to engage the non-specialist participant. A typical example of this would be an inequality problem whose proof requires a tool like the general means inequality or the Cauchy inequality. Such tools can be assumed to be at the disposal of students participating in the MO, but they will likely not be familiar to students whose only applicable knowledge is what they have been exposed to in their regular classrooms.

There are also some more subtle reasons for problems to be unsuited for the MK. Sometimes, problems are suggested that seem at first glance to be quite suitable, but in which the result involves a proof whose structure is of more interest than the result itself. Such problems are intrinsically unsuited to the multiple-choice format. A typical example for such a question is the following.

D) Determine the number of triples of integers solving the equation

Ignoring for a moment the matter of whether this question is sufficiently interesting for a competition, the fact that there are no such integer solutions (each of the squares on the left side of the equation is congruent 1 modulo 3, while the number on the right side of the equation cannot be congruent 2 modulo 3) can conceivably be of interest only if we ask for the proof of the fact. Suggesting possible answers like

is beside the point of the mathematical content of the question being posed. The problem may therefore be usable for a competition requiring justification, but not for the multiple-choice environment of the MK.

Then there is the matter of complexity. This can apply to the problem statement itself or the available derivations of correct answers. If the set-up of a competition problem requires several paragraphs to explain, a competitor will require more time to read and comprehend the question than the time that is available. Typically, an MK paper is composed of 30 problems to be solved in no more than 75 min. This allows an average of 2½ min per question. This includes reading the problem, comprehending the given information, solving the problem and entering the answer on the answer sheet. Even allowing for the fact that a student may be done with the easy problems in less than average time, there is still not much time left over for each of the more difficult problems. Any problem setting composed of more than a few sentences (or with a figure requiring lengthy perusal) will simply take too long to be dealt with appropriately.

Similarly, a problem may simply require too many steps of calculation or logical argument than can reasonably be done in a limited time. This does not mean that the problem is in any way inferior, of course. One and the same problem can often be made shorter or longer, depending on the numbers used. An example is the following question, a 5-point problem taken from the Student paper 2019:

(E) Three different numbers are chosen at random from the set {1, 2, 3, …, 10}. What is the probability that one of them is the average of the other two?

This is not an easy problem for most students. In order to solve this, they must realize that there are $$\left( {\begin{array}{*{20}l} {10} \\ 3 \\ \end{array} } \right) = 120$$ different ways to choose 3 numbers from this 10-element set, and that groups of three numbers with the required property must be of the type x  −  a , x and x  +  a . Since the middle number x can be any of the numbers 2, 3, …, 9, and the number of options for a for each of these are 1, 2, 3, 4, 4, 3, 2 and 1 respectively, there are 1 + 2 + 3 + 4 + 4 + 3 + 2 + 1 = 20 such possibilities, and the probability is therefore $$\frac{20}{{120}} = \frac{1}{6}$$ . This differs markedly from the following two variations of the problem.

(E1) Three different numbers are chosen at random from the set {1, 2, 3, 4}. What is the probability that one of them is the average of the other two?

(E2) Three different numbers are chosen at random from the set {1, 2, 3, 5, 6, 7, 9, 10, 11, 12}. What is the probability that one of them is the average of the other two?

These problems require essentially the same reasoning as the original problem E. Still, problem E1 is obviously much easier to answer than E, as there are only four possible choices for the three numbers and listing them all shows us that the required probability is ½. On the other hand, finding the probability in E2 requires the consideration of many more cases, and is too long for the time allotment of the MK.

With this consideration, it may seem that the avoidance of problems that could be considered ‘too hard’ for the MK should be straight-forward, but this is far from the case. There are many subtle factors at play in the problem-selection process that make the distinction between appropriate hard problems and ‘too-hard’ problems quite difficult in practice.

## 7 Precision is everything. Or is it?

At the MK, a student is faced with many items that are to be dealt with in an average of 2½ min per question. Problem posers must therefore set short and precise tasks, as students will need that time to find their solutions. Unfortunately, a task that is communicated in too abbreviated a fashion may not be clear to the students or, at worst, logically ambiguous. Any attempt at mathematical precision will either require the use of technical terms the students may not have at their disposal, or else the text of the problem will be lengthened. Even worse, this may not even make the problem more understandable. In contrast to MO, however, where students are used to dealing with precise and/or compressed formulations, common notations, and simplifying mathematical symbols, the average participant of the MK cannot be expected to be familiar with this particular ‘language’. The specific vocabulary available for the problem poser also depends on the curricular knowledge of the targeted group of students. In an international group, this will vary from country to country.

The fact that pictures, symbols and signs can be used is of great help for the problem poser, as the following item from Pre-Écolier in 2019 shows:

(F) A mouse and a piece of cheese are in the opposite corners of the board. The mouse can only move as shown by the arrows. In how many ways can the mouse reach the cheese?

Neither the size of the board nor the positions of the two objects nor the possible directions of movement have to be specified due to the available picture and the explanatory addition of the arrows. Such tools make it possible to pack a lot of information into a language-independent format. This is a huge advantage for an international competition, where translation is an issue, but also from the point of view of eliminating the need for unfamiliar technical terms. In the MK 2019, the percentage of items including figures ranges from 37% at the Junior category to 87% at Pre-Écolier. In total, 89 out of a total of 153 items contained figures or pictograms, and unsurprisingly, this number decreases with the increasing age of the participants. Graphic representations are also used as a tool to increase motivation, but to a large extent the intent is for them to replace technical descriptions of mathematical facts and describe geometric objects and operations, such as rotations, reflections, or concrete instances of geometric combinatorics. A further central aspect is that pictograms are used to replace formal mathematical notation and to bypass the use of variables. An example is the following problem from Benjamin 2019:

(G) Bridget folds a square piece of paper twice and subsequently cuts it along the two lines as shown in the picture. How many pieces of paper does she obtain this way?

The depiction of the process of folding and cutting through the use of arrows and dotted lines makes a crucial difference in decreasing the amount of text required to understand the problem.

While technical terms can often be replaced by figures and pictograms, the targeted use of colloquial language is another option that can be considered for this purpose. However, colloquial language can make the wording of a task either shorter or longer, depending on the task. Under certain conditions, a situation might be described colloquially in an abbreviated, and still understandable manner, as a particular situation need not be mathematically embedded. On the other hand, it might actually make the required text longer, if a mathematical fact is to be expressed with sufficient stringency using everyday language, without resorting to mathematically precise, but obscure, terms. An example of this is the following problem from the Junior paper in 2019, where the mathematical term ‘reflection’ is omitted by embedding in a real-world situation:

(H) A barber wants to write the word SHAVE on a board in such a way that a client looking into the mirror reads the word correctly. How should the barber write it on the board?

The problem of possible misinterpretation of colloquial language was addressed by Durand-Guerrier ( 2008 ). In the analysis of a particular logic item of the MK 1994 in France, the author stated that the French version of the task can be solved only if a certain specific word is semantically assigned the exact meaning intended by the problem poser. Of course, since problem posers are aware of this source of conflict, they will generally do their utmost to avoid making any formal mistakes, and to simultaneously provide a balance between colloquial language and precision. Often, this is a matter that greatly depends on the specificities of a language. In an international competition like the MK, this means that the greatest attention in this respect must be paid during the translation of the tasks into the individual languages.

The use of colloquial language and sticking to the everyday meaning of words seems to improve the accessibility of problems at a popular competition in general, but care must be taken to avoid ambiguities. Figures and pictograms can describe mathematical situations, paraphrase definitions and often completely replace lengthy formulations. However, when including figures, two main things should be kept in mind: Firstly, forcing young students to switch between figures and text to get the necessary information in order to solve the task, may decrease a pupil’s performance by increasing the cognitive load on the learner’s limited working memory (Berends & van Lieshout, 2009 ). Secondly, as Carotenuto et al. ( 2021 ) pointed out, the variation in the presentation (e.g., by including figures) significantly changes students’ approaches and answers to word problems in the same context, because the elements included in the presentation of a word problem can have a strong informational component for the students. While the first point further limits the creation of adequate formulations of tasks, especially for younger students, being aware of the second point may even lead to greater variability when creating tasks. Summing up, a compromise between short, precise and manifestly attractive tasks is part of the process of creating problems, even if this is easier to do for certain problems than it is for others.

## 8 Packaging is everything. Or is it?

The internal world of the MK is quite fascinating. In 2019, mice were looking for cheese, mothers cutting cakes, pieces of fruit speaking, kangaroos thinking about their age, giants building sandcastles, barbers attracting customers, ants moving on graphs, and—last, but not least—an endless variation of children dealing with everyday problems like eating chocolate, shaking hands, or throwing balls. On the other hand, some problems simply deal with geometric objects or numbers with certain properties. A problem of the latter, inner-mathematical type, is the following, from the Student paper 2019:

(I) A positive integer n is called good if its largest divisor (excluding n ) is equal to $$n - 6$$ . How many good positive integers are there?

Problem posers often try to use the daily-life experiences of the younger students pragmatically in order to have a contextually broad variation of similar mathematical content. This is necessary, as their mathematical knowledge is still limited. In particular, this can be seen in Pre-Écolier, where only one of 2019’s tasks is inner-mathematical. At the other end of the age spectrum, there are barely any hard tasks which are not inner-mathematical. An explanation may be the fact that the items are already hard from a mathematical point of view and therefore intentionally stripped of any additional time-consuming complexity that would be introduced by requiring the translation of some real-world context into mathematics.

Some authors have a great love of packing mathematical problems into stories, literary contexts, and even fairy tales (see Kašuba, 2017 ). Their argument is that these kinds of problems improve the motivation of the students, especially younger ones. In return, it can sometimes be the case that a context will come across as being somehow artificial or too unrealistic. Studies show a positive effect on motivation in providing real-world and modelling problems, in particular for weak students (see Maaß & Mischo, 2012 ; Pongsakdi et al., 2019 ), which is of course a major goal of a popular competition. On the other hand, Rellensmann and Schukajlow ( 2017 ) found that ninth grade students experienced the same levels of enjoyment and boredom when solving problems with and without a connection to reality. It should be pointed out that motivation and performance seem to be related: By means of an interventional study, Pongsakdi et al. ( 2019 ) found that the problem-solving performance of students in grades four and six, working on a teacher’s innovative self-created challenging word problems, increases when compared to students faced with typical word problems, when the initial motivational level of the students was already high. Furthermore, there is a correlation between text comprehension and performance both for easy and difficult tasks (Plath, 2020 ; Pongsakdi et al., 2020 ). Plath ( 2020 ) also showed that performance generally decreases when the linguistic complexity of a word problem is increased and the mathematical complexity is simultaneously left unchanged, and that half of the time spent solving a word problem is spent on understanding the setting and stripping it down to its mathematical content.

The research findings stated above can be transferred to the MK only to a certain extent, since that competition contains a very special subset of word problems, namely word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies, due to the use of the classification of word problems by Verschaffel et al. ( 2020 ). There does, however, seem to be some evidence suggesting that real-world settings can provide positive motivational effects for many weaker or younger students. On the other hand, it may actually decrease the probability of solving the problem due to the additional challenges posed by the requirement of reading information from additional (and possibly unnecessarily complex) contexts. The question of the extent to which the arithmetical and mathematical problem-solving skills and/or text comprehension should be the main focus seems to have no ultimately correct answer, and is addressed annually by the selection groups. A main challenge for the community is to choose and select a good mix of problems for the MK, where all the various requirements are taken into account.

## 9 Conclusion

In this paper, we have tried to discuss all relevant properties and aspects of potential competition problems, including the following: differences from textbook problems, content of problems, the crucial role of distractors, the desire for problems to be both easy and interesting, the range of difficulty, the precision and the packaging. Furthermore, we embedded the practitioner’s view of creating problems in research on problem posing and word-problem solving, thereby addressing an important need to connect the Competition Movement and the mathematics education community.

After a close look at some specific problems from the classroom and the world of the competitions in Sect.  2 , the de facto competition curriculum was explored in Sect.  3 . In Sects.  4 – 6 the creation of high-quality problems and distractors for the MK was considered, with a special focus on the theory of problem posing. In Sect.  7 , details of the formulation of multiple-choice competition problems and the role of figures were discussed. Finally, in Sect.  8 , we highlighted the variation of design within the tasks of the MK and identified connections to current research on word-problem solving. This last point could serve as a starting point for more detailed investigations, for instance when focusing on students’ motivation when solving problems of the MK, and it could contribute to current discussions of affective factors in solving complex word problems.

Besides being aware of all these aspects, experience seems to be important in order to create high-quality problems for the MK, as becomes apparent in numerous instances within the paper. Lee ( 2020 ) pointed out that “[i]t would be beneficial to conduct problem posing studies with mathematics experts because it can not only provide students with experiences characterizing mathematics experts’ thinking but also assist students to learn the ways of thinking while facing problems” (Lee, 2020 , p. 12). Despite the fact that the limitations of the study are quite clear as it derived from the experiences of both authors, we are convinced that our analysis serves as a starting point for a discourse on essential aspects of problems for popular mathematics competitions, as well as for empirically investigating factors that allow rich and productive problem posing for popular competitions. Such investigations could give a better understanding of the nature of problem posing itself and hence enrich the perspective of problem posing as a research goal (Cai & Leikin, 2020 ).

Ultimately, it must be remembered that the whole point of the exercise is to motivate students and to give them a joyful experience, new insights and a different perspective on the fascinating and versatile world of mathematics. This may sometimes require some difficult decisions, but careful consideration of all these nuances will certainly result in competitions that participants can ultimately profit from and enjoy.

The adults of the ‘Competition Movement’ consist of competition designers and organisers and experts in problem posing.

The author L. Donner was formerly known as L. Andritsch.

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Geretschläger, R., Donner, L. Writing and choosing problems for a popular high school mathematics competition. ZDM Mathematics Education 54 , 971–982 (2022). https://doi.org/10.1007/s11858-022-01351-9

Accepted : 09 March 2022

Published : 29 March 2022

Issue Date : October 2022

DOI : https://doi.org/10.1007/s11858-022-01351-9

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Original research article, mathematical problem-solving through cooperative learning—the importance of peer acceptance and friendships.

• 1 Department of Education, Uppsala University, Uppsala, Sweden
• 2 Department of Education, Culture and Communication, Malardalen University, Vasteras, Sweden
• 3 School of Natural Sciences, Technology and Environmental Studies, Sodertorn University, Huddinge, Sweden
• 4 Faculty of Education, Gothenburg University, Gothenburg, Sweden

Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional approaches supporting student learning in this area. This study aims to contribute to previous research by studying the effects of an instructional approach of cooperative learning on students’ mathematical problem-solving in heterogeneous classrooms in grade five, in which students with special needs are educated alongside with their peers. The intervention combined a cooperative learning approach with instruction in problem-solving strategies including mathematical models of multiplication/division, proportionality, and geometry. The teachers in the experimental group received training in cooperative learning and mathematical problem-solving, and implemented the intervention for 15 weeks. The teachers in the control group received training in mathematical problem-solving and provided instruction as they would usually. Students (269 in the intervention and 312 in the control group) participated in tests of mathematical problem-solving in the areas of multiplication/division, proportionality, and geometry before and after the intervention. The results revealed significant effects of the intervention on student performance in overall problem-solving and problem-solving in geometry. The students who received higher scores on social acceptance and friendships for the pre-test also received higher scores on the selected tests of mathematical problem-solving. Thus, the cooperative learning approach may lead to gains in mathematical problem-solving in heterogeneous classrooms, but social acceptance and friendships may also greatly impact students’ results.

## Introduction

The research on instruction in mathematical problem-solving has progressed considerably during recent decades. Yet, there is still a need to advance our knowledge on how teachers can support their students in carrying out this complex activity ( Lester and Cai, 2016 ). Results from the Program for International Student Assessment (PISA) show that only 53% of students from the participating countries could solve problems requiring more than direct inference and using representations from different information sources ( OECD, 2019 ). In addition, OECD (2019) reported a large variation in achievement with regard to students’ diverse backgrounds. Thus, there is a need for instructional approaches to promote students’ problem-solving in mathematics, especially in heterogeneous classrooms in which students with diverse backgrounds and needs are educated together. Small group instructional approaches have been suggested as important to promote learning of low-achieving students and students with special needs ( Kunsch et al., 2007 ). One such approach is cooperative learning (CL), which involves structured collaboration in heterogeneous groups, guided by five principles to enhance group cohesion ( Johnson et al., 1993 ; Johnson et al., 2009 ; Gillies, 2016 ). While CL has been well-researched in whole classroom approaches ( Capar and Tarim, 2015 ), few studies of the approach exist with regard to students with special educational needs (SEN; McMaster and Fuchs, 2002 ). This study contributes to previous research by studying the effects of the CL approach on students’ mathematical problem-solving in heterogeneous classrooms, in which students with special needs are educated alongside with their peers.

Group collaboration through the CL approach is structured in accordance with five principles of collaboration: positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing ( Johnson et al., 1993 ). First, the group tasks need to be structured so that all group members feel dependent on each other in the completion of the task, thus promoting positive interdependence. Second, for individual accountability, the teacher needs to assure that each group member feels responsible for his or her share of work, by providing opportunities for individual reports or evaluations. Third, the students need explicit instruction in social skills that are necessary for collaboration. Fourth, the tasks and seat arrangements should be designed to promote interaction among group members. Fifth, time needs to be allocated to group processing, through which group members can evaluate their collaborative work to plan future actions. Using these principles for cooperation leads to gains in mathematics, according to Capar and Tarim (2015) , who conducted a meta-analysis on studies of cooperative learning and mathematics, and found an increase of .59 on students’ mathematics achievement scores in general. However, the number of reviewed studies was limited, and researchers suggested a need for more research. In the current study, we focused on the effect of CL approach in a specific area of mathematics: problem-solving.

Mathematical problem-solving is a central area of mathematics instruction, constituting an important part of preparing students to function in modern society ( Gravemeijer et al., 2017 ). In fact, problem-solving instruction creates opportunities for students to apply their knowledge of mathematical concepts, integrate and connect isolated pieces of mathematical knowledge, and attain a deeper conceptual understanding of mathematics as a subject ( Lester and Cai, 2016 ). Some researchers suggest that mathematics itself is a science of problem-solving and of developing theories and methods for problem-solving ( Hamilton, 2007 ; Davydov, 2008 ).

Problem-solving processes have been studied from different perspectives ( Lesh and Zawojewski, 2007 ). Problem-solving heuristics Pólya, (1948) has largely influenced our perceptions of problem-solving, including four principles: understanding the problem, devising a plan, carrying out the plan, and looking back and reflecting upon the suggested solution. Schoenfield, (2016) suggested the use of specific problem-solving strategies for different types of problems, which take into consideration metacognitive processes and students’ beliefs about problem-solving. Further, models and modelling perspectives on mathematics ( Lesh and Doerr, 2003 ; Lesh and Zawojewski, 2007 ) emphasize the importance of engaging students in model-eliciting activities in which problem situations are interpreted mathematically, as students make connections between problem information and knowledge of mathematical operations, patterns, and rules ( Mousoulides et al., 2010 ; Stohlmann and Albarracín, 2016 ).

Not all students, however, find it easy to solve complex mathematical problems. Students may experience difficulties in identifying solution-relevant elements in a problem or visualizing appropriate solution to a problem situation. Furthermore, students may need help recognizing the underlying model in problems. For example, in two studies by Degrande et al. (2016) , students in grades four to six were presented with mathematical problems in the context of proportional reasoning. The authors found that the students, when presented with a word problem, could not identify an underlying model, but rather focused on superficial characteristics of the problem. Although the students in the study showed more success when presented with a problem formulated in symbols, the authors pointed out a need for activities that help students distinguish between different proportional problem types. Furthermore, students exhibiting specific learning difficulties may need additional support in both general problem-solving strategies ( Lein et al., 2020 ; Montague et al., 2014 ) and specific strategies pertaining to underlying models in problems. The CL intervention in the present study focused on supporting students in problem-solving, through instruction in problem-solving principles ( Pólya, 1948 ), specifically applied to three models of mathematical problem-solving—multiplication/division, geometry, and proportionality.

Students’ problem-solving may be enhanced through participation in small group discussions. In a small group setting, all the students have the opportunity to explain their solutions, clarify their thinking, and enhance understanding of a problem at hand ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ). In fact, small group instruction promotes students’ learning in mathematics by providing students with opportunities to use language for reasoning and conceptual understanding ( Mercer and Sams, 2006 ), to exchange different representations of the problem at hand ( Fujita et al., 2019 ), and to become aware of and understand groupmates’ perspectives in thinking ( Kazak et al., 2015 ). These opportunities for learning are created through dialogic spaces characterized by openness to each other’s perspectives and solutions to mathematical problems ( Wegerif, 2011 ).

However, group collaboration is not only associated with positive experiences. In fact, studies show that some students may not be given equal opportunities to voice their opinions, due to academic status differences ( Langer-Osuna, 2016 ). Indeed, problem-solvers struggling with complex tasks may experience negative emotions, leading to uncertainty of not knowing the definite answer, which places demands on peer support ( Jordan and McDaniel, 2014 ; Hannula, 2015 ). Thus, especially in heterogeneous groups, students may need additional support to promote group interaction. Therefore, in this study, we used a cooperative learning approach, which, in contrast to collaborative learning approaches, puts greater focus on supporting group cohesion through instruction in social skills and time for reflection on group work ( Davidson and Major, 2014 ).

Although cooperative learning approach is intended to promote cohesion and peer acceptance in heterogeneous groups ( Rzoska and Ward, 1991 ), previous studies indicate that challenges in group dynamics may lead to unequal participation ( Mulryan, 1992 ; Cohen, 1994 ). Peer-learning behaviours may impact students’ problem-solving ( Hwang and Hu, 2013 ) and working in groups with peers who are seen as friends may enhance students’ motivation to learn mathematics ( Deacon and Edwards, 2012 ). With the importance of peer support in mind, this study set out to investigate whether the results of the intervention using the CL approach are associated with students’ peer acceptance and friendships.

## The Present Study

In previous research, the CL approach has shown to be a promising approach in teaching and learning mathematics ( Capar and Tarim, 2015 ), but fewer studies have been conducted in whole-class approaches in general and students with SEN in particular ( McMaster and Fuchs, 2002 ). This study aims to contribute to previous research by investigating the effect of CL intervention on students’ mathematical problem-solving in grade 5. With regard to the complexity of mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach in this study was combined with problem-solving principles pertaining to three underlying models of problem-solving—multiplication/division, geometry, and proportionality. Furthermore, considering the importance of peer support in problem-solving in small groups ( Mulryan, 1992 ; Cohen, 1994 ; Hwang and Hu, 2013 ), the study investigated how peer acceptance and friendships were associated with the effect of the CL approach on students’ problem-solving abilities. The study aimed to find answers to the following research questions:

a) What is the effect of CL approach on students’ problem-solving in mathematics?

b) Are social acceptance and friendship associated with the effect of CL on students’ problem-solving in mathematics?

## Participants

The participants were 958 students in grade 5 and their teachers. According to power analyses prior to the start of the study, 1,020 students and 51 classes were required, with an expected effect size of 0.30 and power of 80%, provided that there are 20 students per class and intraclass correlation is 0.10. An invitation to participate in the project was sent to teachers in five municipalities via e-mail. Furthermore, the information was posted on the website of Uppsala university and distributed via Facebook interest groups. As shown in Figure 1 , teachers of 1,165 students agreed to participate in the study, but informed consent was obtained only for 958 students (463 in the intervention and 495 in the control group). Further attrition occurred at pre- and post-measurement, resulting in 581 students’ tests as a basis for analyses (269 in the intervention and 312 in the control group). Fewer students (n = 493) were finally included in the analyses of the association of students’ social acceptance and friendships and the effect of CL on students’ mathematical problem-solving (219 in the intervention and 274 in the control group). The reasons for attrition included teacher drop out due to sick leave or personal circumstances (two teachers in the control group and five teachers in the intervention group). Furthermore, some students were sick on the day of data collection and some teachers did not send the test results to the researchers.

FIGURE 1 . Flow chart for participants included in data collection and data analysis.

As seen in Table 1 , classes in both intervention and control groups included 27 students on average. For 75% of the classes, there were 33–36% of students with SEN. In Sweden, no formal medical diagnosis is required for the identification of students with SEN. It is teachers and school welfare teams who decide students’ need for extra adaptations or special support ( Swedish National Educational Agency, 2014 ). The information on individual students’ type of SEN could not be obtained due to regulations on the protection of information about individuals ( SFS 2009 ). Therefore, the information on the number of students with SEN on class level was obtained through teacher reports.

TABLE 1 . Background characteristics of classes and teachers in intervention and control groups.

## Intervention

The intervention using the CL approach lasted for 15 weeks and the teachers worked with the CL approach three to four lessons per week. First, the teachers participated in two-days training on the CL approach, using an especially elaborated CL manual ( Klang et al., 2018 ). The training focused on the five principles of the CL approach (positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing). Following the training, the teachers introduced the CL approach in their classes and focused on group-building activities for 7 weeks. Then, 2 days of training were provided to teachers, in which the CL approach was embedded in activities in mathematical problem-solving and reading comprehension. Educational materials containing mathematical problems in the areas of multiplication and division, geometry, and proportionality were distributed to the teachers ( Karlsson and Kilborn, 2018a ). In addition to the specific problems, adapted for the CL approach, the educational materials contained guidance for the teachers, in which problem-solving principles ( Pólya, 1948 ) were presented as steps in problem-solving. Following the training, the teachers applied the CL approach in mathematical problem-solving lessons for 8 weeks.

Solving a problem is a matter of goal-oriented reasoning, starting from the understanding of the problem to devising its solution by using known mathematical models. This presupposes that the current problem is chosen from a known context ( Stillman et al., 2008 ; Zawojewski, 2010 ). This differs from the problem-solving of the textbooks, which is based on an aim to train already known formulas and procedures ( Hamilton, 2007 ). Moreover, it is important that students learn modelling according to their current abilities and conditions ( Russel, 1991 ).

In order to create similar conditions in the experiment group and the control group, the teachers were supposed to use the same educational material ( Karlsson and Kilborn, 2018a ; Karlsson and Kilborn, 2018b ), written in light of the specified view of problem-solving. The educational material is divided into three areas—multiplication/division, geometry, and proportionality—and begins with a short teachers’ guide, where a view of problem solving is presented, which is based on the work of Polya (1948) and Lester and Cai (2016) . The tasks are constructed in such a way that conceptual knowledge was in focus, not formulas and procedural knowledge.

## Implementation of the Intervention

To ensure the implementation of the intervention, the researchers visited each teachers’ classroom twice during the two phases of the intervention period, as described above. During each visit, the researchers observed the lesson, using a checklist comprising the five principles of the CL approach. After the lesson, the researchers gave written and oral feedback to each teacher. As seen in Table 1 , in 18 of the 23 classes, the teachers implemented the intervention in accordance with the principles of CL. In addition, the teachers were asked to report on the use of the CL approach in their teaching and the use of problem-solving activities embedding CL during the intervention period. As shown in Table 1 , teachers in only 11 of 23 classes reported using the CL approach and problem-solving activities embedded in the CL approach at least once a week.

## Control Group

The teachers in the control group received 2 days of instruction in enhancing students’ problem-solving and reading comprehension. The teachers were also supported with educational materials including mathematical problems Karlsson and Kilborn (2018b) and problem-solving principles ( Pólya, 1948 ). However, none of the activities during training or in educational materials included the CL approach. As seen in Table 1 , only 10 of 25 teachers reported devoting at least one lesson per week to mathematical problem-solving.

## Tests of Mathematical Problem-Solving

Tests of mathematical problem-solving were administered before and after the intervention, which lasted for 15 weeks. The tests were focused on the models of multiplication/division, geometry, and proportionality. The three models were chosen based on the syllabus of the subject of mathematics in grades 4 to 6 in the Swedish National Curriculum ( Swedish National Educational Agency, 2018 ). In addition, the intention was to create a variation of types of problems to solve. For each of these three models, there were two tests, a pre-test and a post-test. Each test contained three tasks with increasing difficulty ( Supplementary Appendix SA ).

The tests of multiplication and division (Ma1) were chosen from different contexts and began with a one-step problem, while the following two tasks were multi-step problems. Concerning multiplication, many students in grade 5 still understand multiplication as repeated addition, causing significant problems, as this conception is not applicable to multiplication beyond natural numbers ( Verschaffel et al., 2007 ). This might be a hindrance in developing multiplicative reasoning ( Barmby et al., 2009 ). The multi-step problems in this study were constructed to support the students in multiplicative reasoning.

Concerning the geometry tests (Ma2), it was important to consider a paradigm shift concerning geometry in education that occurred in the mid-20th century, when strict Euclidean geometry gave way to other aspects of geometry like symmetry, transformation, and patterns. van Hiele (1986) prepared a new taxonomy for geometry in five steps, from a visual to a logical level. Therefore, in the tests there was a focus on properties of quadrangles and triangles, and how to determine areas by reorganising figures into new patterns. This means that structure was more important than formulas.

The construction of tests of proportionality (M3) was more complicated. Firstly, tasks on proportionality can be found in many different contexts, such as prescriptions, scales, speeds, discounts, interest, etc. Secondly, the mathematical model is complex and requires good knowledge of rational numbers and ratios ( Lesh et al., 1988 ). It also requires a developed view of multiplication, useful in operations with real numbers, not only as repeated addition, an operation limited to natural numbers ( Lybeck, 1981 ; Degrande et al., 2016 ). A linear structure of multiplication as repeated addition leads to limitations in terms of generalization and development of the concept of multiplication. This became evident in a study carried out in a Swedish context ( Karlsson and Kilborn, 2018c ). Proportionality can be expressed as a/b = c/d or as a/b = k. The latter can also be expressed as a = b∙k, where k is a constant that determines the relationship between a and b. Common examples of k are speed (km/h), scale, and interest (%). An important pre-knowledge in order to deal with proportions is to master fractions as equivalence classes like 1/3 = 2/6 = 3/9 = 4/12 = 5/15 = 6/18 = 7/21 = 8/24 … ( Karlsson and Kilborn, 2020 ). It was important to take all these aspects into account when constructing and assessing the solutions of the tasks.

The tests were graded by an experienced teacher of mathematics (4 th author) and two students in their final year of teacher training. Prior to grading, acceptable levels of inter-rater reliability were achieved by independent rating of students’ solutions and discussions in which differences between the graders were resolved. Each student response was to be assigned one point when it contained a correct answer and two points when the student provided argumentation for the correct answer and elaborated on explanation of his or her solution. The assessment was thus based on quality aspects with a focus on conceptual knowledge. As each subtest contained three questions, it generated three student solutions. So, scores for each subtest ranged from 0 to 6 points and for the total scores from 0 to 18 points. To ascertain that pre- and post-tests were equivalent in degree of difficulty, the tests were administered to an additional sample of 169 students in grade 5. Test for each model was conducted separately, as students participated in pre- and post-test for each model during the same lesson. The order of tests was switched for half of the students in order to avoid the effect of the order in which the pre- and post-tests were presented. Correlation between students’ performance on pre- and post-test was .39 ( p < 0.000) for tests of multiplication/division; .48 ( p < 0.000) for tests of geometry; and .56 ( p < 0.000) for tests of proportionality. Thus, the degree of difficulty may have differed between pre- and post-test.

## Measures of Peer Acceptance and Friendships

To investigate students’ peer acceptance and friendships, peer nominations rated pre- and post-intervention were used. Students were asked to nominate peers who they preferred to work in groups with and who they preferred to be friends with. Negative peer nominations were avoided due to ethical considerations raised by teachers and parents ( Child and Nind, 2013 ). Unlimited nominations were used, as these are considered to have high ecological validity ( Cillessen and Marks, 2017 ). Peer nominations were used as a measure of social acceptance, and reciprocated nominations were used as a measure of friendship. The number of nominations for each student were aggregated and divided by the number of nominators to create a proportion of nominations for each student ( Velásquez et al., 2013 ).

## Statistical Analyses

Multilevel regression analyses were conducted in R, lme4 package Bates et al. (2015) to account for nestedness in the data. Students’ classroom belonging was considered as a level 2 variable. First, we used a model in which students’ results on tests of problem-solving were studied as a function of time (pre- and post) and group belonging (intervention and control group). Second, the same model was applied to subgroups of students who performed above and below median at pre-test, to explore whether the CL intervention had a differential effect on student performance. In this second model, the results for subgroups of students could not be obtained for geometry tests for subgroup below median and for tests of proportionality for subgroup above median. A possible reason for this must have been the skewed distribution of the students in these subgroups. Therefore, another model was applied that investigated students’ performances in math at both pre- and post-test as a function of group belonging. Third, the students’ scores on social acceptance and friendships were added as an interaction term to the first model. In our previous study, students’ social acceptance changed as a result of the same CL intervention ( Klang et al., 2020 ).

The assumptions for the multilevel regression were assured during the analyses ( Snijders and Bosker, 2012 ). The assumption of normality of residuals were met, as controlled by visual inspection of quantile-quantile plots. For subgroups, however, the plotted residuals deviated somewhat from the straight line. The number of outliers, which had a studentized residual value greater than ±3, varied from 0 to 5, but none of the outliers had a Cook’s distance value larger than 1. The assumption of multicollinearity was met, as the variance inflation factors (VIF) did not exceed a value of 10. Before the analyses, the cases with missing data were deleted listwise.

## What Is the Effect of the CL Approach on Students’ Problem-Solving in Mathematics?

As seen in the regression coefficients in Table 2 , the CL intervention had a significant effect on students’ mathematical problem-solving total scores and students’ scores in problem solving in geometry (Ma2). Judging by mean values, students in the intervention group appeared to have low scores on problem-solving in geometry but reached the levels of problem-solving of the control group by the end of the intervention. The intervention did not have a significant effect on students’ performance in problem-solving related to models of multiplication/division and proportionality.

TABLE 2 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving.

The question is, however, whether CL intervention affected students with different pre-test scores differently. Table 2 includes the regression coefficients for subgroups of students who performed below and above median at pre-test. As seen in the table, the CL approach did not have a significant effect on students’ problem-solving, when the sample was divided into these subgroups. A small negative effect was found for intervention group in comparison to control group, but confidence intervals (CI) for the effect indicate that it was not significant.

## Is Social Acceptance and Friendships Associated With the Effect of CL on Students’ Problem-Solving in Mathematics?

As seen in Table 3 , students’ peer acceptance and friendship at pre-test were significantly associated with the effect of the CL approach on students’ mathematical problem-solving scores. Changes in students’ peer acceptance and friendships were not significantly associated with the effect of the CL approach on students’ mathematical problem-solving. Consequently, it can be concluded that being nominated by one’s peers and having friends at the start of the intervention may be an important factor when participation in group work, structured in accordance with the CL approach, leads to gains in mathematical problem-solving.

TABLE 3 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving, including scores of social acceptance and friendship in the model.

In light of the limited number of studies on the effects of CL on students’ problem-solving in whole classrooms ( Capar and Tarim, 2015 ), and for students with SEN in particular ( McMaster and Fuchs, 2002 ), this study sought to investigate whether the CL approach embedded in problem-solving activities has an effect on students’ problem-solving in heterogeneous classrooms. The need for the study was justified by the challenge of providing equitable mathematics instruction to heterogeneous student populations ( OECD, 2019 ). Small group instructional approaches as CL are considered as promising approaches in this regard ( Kunsch et al., 2007 ). The results showed a significant effect of the CL approach on students’ problem-solving in geometry and total problem-solving scores. In addition, with regard to the importance of peer support in problem-solving ( Deacon and Edwards, 2012 ; Hwang and Hu, 2013 ), the study explored whether the effect of CL on students’ problem-solving was associated with students’ social acceptance and friendships. The results showed that students’ peer acceptance and friendships at pre-test were significantly associated with the effect of the CL approach, while change in students’ peer acceptance and friendships from pre- to post-test was not.

The results of the study confirm previous research on the effect of the CL approach on students’ mathematical achievement ( Capar and Tarim, 2015 ). The specific contribution of the study is that it was conducted in classrooms, 75% of which were composed of 33–36% of students with SEN. Thus, while a previous review revealed inconclusive findings on the effects of CL on student achievement ( McMaster and Fuchs, 2002 ), the current study adds to the evidence of the effect of the CL approach in heterogeneous classrooms, in which students with special needs are educated alongside with their peers. In a small group setting, the students have opportunities to discuss their ideas of solutions to the problem at hand, providing explanations and clarifications, thus enhancing their understanding of problem-solving ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ).

In this study, in accordance with previous research on mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach was combined with training in problem-solving principles Pólya (1948) and educational materials, providing support in instruction in underlying mathematical models. The intention of the study was to provide evidence for the effectiveness of the CL approach above instruction in problem-solving, as problem-solving materials were accessible to teachers of both the intervention and control groups. However, due to implementation challenges, not all teachers in the intervention and control groups reported using educational materials and training as expected. Thus, it is not possible to draw conclusions of the effectiveness of the CL approach alone. However, in everyday classroom instruction it may be difficult to separate the content of instruction from the activities that are used to mediate this content ( Doerr and Tripp, 1999 ; Gravemeijer, 1999 ).

Furthermore, for successful instruction in mathematical problem-solving, scaffolding for content needs to be combined with scaffolding for dialogue ( Kazak et al., 2015 ). From a dialogical perspective ( Wegerif, 2011 ), students may need scaffolding in new ways of thinking, involving questioning their understandings and providing arguments for their solutions, in order to create dialogic spaces in which different solutions are voiced and negotiated. In this study, small group instruction through CL approach aimed to support discussions in small groups, but the study relies solely on quantitative measures of students’ mathematical performance. Video-recordings of students’ discussions may have yielded important insights into the dialogic relationships that arose in group discussions.

Despite the positive findings of the CL approach on students’ problem-solving, it is important to note that the intervention did not have an effect on students’ problem-solving pertaining to models of multiplication/division and proportionality. Although CL is assumed to be a promising instructional approach, the number of studies on its effect on students’ mathematical achievement is still limited ( Capar and Tarim, 2015 ). Thus, further research is needed on how CL intervention can be designed to promote students’ problem-solving in other areas of mathematics.

The results of this study show that the effect of the CL intervention on students’ problem-solving was associated with students’ initial scores of social acceptance and friendships. Thus, it is possible to assume that students who were popular among their classmates and had friends at the start of the intervention also made greater gains in mathematical problem-solving as a result of the CL intervention. This finding is in line with Deacon and Edwards’ study of the importance of friendships for students’ motivation to learn mathematics in small groups ( Deacon and Edwards, 2012 ). However, the effect of the CL intervention was not associated with change in students’ social acceptance and friendship scores. These results indicate that students who were nominated by a greater number of students and who received a greater number of friends did not benefit to a great extent from the CL intervention. With regard to previously reported inequalities in cooperation in heterogeneous groups ( Cohen, 1994 ; Mulryan, 1992 ; Langer Osuna, 2016 ) and the importance of peer behaviours for problem-solving ( Hwang and Hu, 2013 ), teachers should consider creating inclusive norms and supportive peer relationships when using the CL approach. The demands of solving complex problems may create negative emotions and uncertainty ( Hannula, 2015 ; Jordan and McDaniel, 2014 ), and peer support may be essential in such situations.

## Limitations

The conclusions from the study must be interpreted with caution, due to a number of limitations. First, due to the regulation of protection of individuals ( SFS 2009 ), the researchers could not get information on type of SEN for individual students, which limited the possibilities of the study for investigating the effects of the CL approach for these students. Second, not all teachers in the intervention group implemented the CL approach embedded in problem-solving activities and not all teachers in the control group reported using educational materials on problem-solving. The insufficient levels of implementation pose a significant challenge to the internal validity of the study. Third, the additional investigation to explore the equivalence in difficulty between pre- and post-test, including 169 students, revealed weak to moderate correlation in students’ performance scores, which may indicate challenges to the internal validity of the study.

## Implications

The results of the study have some implications for practice. Based on the results of the significant effect of the CL intervention on students’ problem-solving, the CL approach appears to be a promising instructional approach in promoting students’ problem-solving. However, as the results of the CL approach were not significant for all subtests of problem-solving, and due to insufficient levels of implementation, it is not possible to conclude on the importance of the CL intervention for students’ problem-solving. Furthermore, it appears to be important to create opportunities for peer contacts and friendships when the CL approach is used in mathematical problem-solving activities.

## Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

## Ethics Statement

The studies involving human participants were reviewed and approved by the Uppsala Ethical Regional Committee, Dnr. 2017/372. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.

## Author Contributions

NiK was responsible for the project, and participated in data collection and data analyses. NaK and WK were responsible for intervention with special focus on the educational materials and tests in mathematical problem-solving. PE participated in the planning of the study and the data analyses, including coordinating analyses of students’ tests. MK participated in the designing and planning the study as well as data collection and data analyses.

The project was funded by the Swedish Research Council under Grant 2016-04,679.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## Acknowledgments

We would like to express our gratitude to teachers who participated in the project.

## Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feduc.2021.710296/full#supplementary-material

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Klang, N., Olsson, I., Wilder, J., Lindqvist, G., Fohlin, N., and Nilholm, C. (2020). A cooperative learning intervention to promote social inclusion in heterogeneous classrooms. Front. Psychol. 11, 586489. doi:10.3389/fpsyg.2020.586489

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Kunsch, C. A., Jitendra, A. K., and Sood, S. (2007). The effects of peer-mediated instruction in mathematics for students with learning problems: A research synthesis. Learn. Disabil Res Pract 22 (1), 1–12. doi:10.1111/j.1540-5826.2007.00226.x

Langer-Osuna, J. M. (2016). The social construction of authority among peers and its implications for collaborative mathematics problem solving. Math. Thinking Learn. 18 (2), 107–124. doi:10.1080/10986065.2016.1148529

Lein, A. E., Jitendra, A. K., and Harwell, M. R. (2020). Effectiveness of mathematical word problem solving interventions for students with learning disabilities and/or mathematics difficulties: A meta-analysis. J. Educ. Psychol. 112 (7), 1388–1408. doi:10.1037/edu0000453

Lesh, R., and Doerr, H. (2003). Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning and Teaching . Mahwah, NJ: Erlbaum .

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Zawojewski, J. (2010). Problem Solving versus Modeling. In R. Lesch, P. Galbraith, C. R. Haines, and A. Hurford (red.), Modelling student’s mathematical modelling competencies: ICTMA , p. 237–243. New York, NY: Springer .doi:10.1007/978-1-4419-0561-1_20

Keywords: cooperative learning, mathematical problem-solving, intervention, heterogeneous classrooms, hierarchical linear regression analysis

Citation: Klang N, Karlsson N, Kilborn W, Eriksson P and Karlberg M (2021) Mathematical Problem-Solving Through Cooperative Learning—The Importance of Peer Acceptance and Friendships. Front. Educ. 6:710296. doi: 10.3389/feduc.2021.710296

Received: 15 May 2021; Accepted: 09 August 2021; Published: 24 August 2021.

Reviewed by:

Copyright © 2021 Klang, Karlsson, Kilborn, Eriksson and Karlberg. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Nina Klang, [email protected]

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## 10 Strategies for Problem Solving in Math

Jessica Kaminski

November 20, 2023

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

## What Are Problem Solving Strategies in Math?

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

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

## Strategies for Problem-solving in Math

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

## Understand the Problem

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

Math for Kids

## Guess and Check

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

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

## Work It Out

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

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

## Work Backwards

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

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

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

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

## Find a Pattern

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

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

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

## Draw a Picture or Diagram

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

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

## Trial and Error Method

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

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

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

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

Check out the Printable Math Worksheets for Your Kids!

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

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

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• Our program for 1st to 8th grade students is aligned with School Math Curriculum.

After-School Math Programs Our program for 1st to 8th grade students is aligned with School Math Curriculum.

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## Module 1: Problem Solving Strategies

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

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

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

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

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

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

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

• Our Mission

There has been an error with the video.

## Building Problem-Solving Skills Through ‘Speed Dating’

When students solve each other’s problems, they gain confidence in their own skills and witness the power of collaboration in real time.

When English language arts teacher Cathleen Beachboard was trying to come up with new ways of encouraging her students at Fauquier High School in Warrenton, Virginia, to be solution-oriented, she noticed that while kids often get stuck on their own problems, they are quick to find solutions to other people’s. Her husband jokingly said, ”Wouldn’t it be cool if you could just speed-date problems?” And the problem speed-dating activity was born. Beachboard has used this quick and simple strategy in her classes ever since.

“Today we were working on argumentative writing. But you could use this in any scenario, in any classroom, for any stuck point that the kids are having with any piece of content,” she says.

In addition to hearing multiple ideas for solutions to their problems, kids get to know each other—and learn to see each other as experts in the room.

“Problem speed dating ultimately teaches them that as a society or as a person, you’re not alone,” Beachboard says. “And reaching out to others is important because that’s where the magic happens. It happens when we use each other and we use the strength that every person has.”

Learn more about Beachboard’s teaching strategies from her posts for Edutopia , or check out her book, The School of Hope .

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## Older Adults and Balance Problems

Have you ever felt dizzy, lightheaded, or as if the room were spinning around you? These can be troublesome sensations. If the feeling happens often, it could be a sign of a balance problem.

## Causes of balance problems

Symptoms of balance disorders, treatments for balance problems and disorders, coping with a balance disorder.

Many older adults experience problems with balance and dizziness. Problems can be caused by certain medications, balance disorders, or other medical conditions. Balance problems are one reason older people fall. Maintaining good balance as you age and learning about fall prevention can help you get around, stay independent, and carry out daily activities.

People are more likely to have problems with balance as they grow older. In some cases, you can help reduce your risk for certain balance problems, but problems often can start suddenly and without obvious cause.

Balance problems can be caused by certain medications or medical conditions. The list below covers some common causes of balance problems.

• Medications. Check with your doctor if you notice balance problems while taking certain medications. Ask if other medications can be used instead, if the dosage can be safely reduced, or if there are other ways to reduce unwanted side effects.
• Inner ear problems. A part of the inner ear called the labyrinth is responsible for balance. When the labyrinth becomes inflamed, a condition called labyrinthitis occurs, causing vertigo and imbalance. Certain ear diseases and infections can lead to labyrinthitis.
• Alcohol. Alcohol in the blood can also cause dizziness and balance problems by affecting how the inner ear works.
• Other medical conditions. Certain conditions, such as diabetes, heart disease, stroke , or problems with your vision, thyroid, nerves, or blood vessels can cause dizziness and other balance problems.

Visit the NIH National Institute on Deafness and Other Communication Disorders website for more information on specific balance disorders .

If you have a balance disorder, you might experience symptoms such as:

• Dizziness or vertigo (a spinning sensation)
• Falling or feeling as if you are going to fall
• Staggering when you try to walk
• Lightheadedness, faintness, or a floating sensation
• Blurred vision
• Confusion or disorientation

Other symptoms might include nausea and vomiting; diarrhea; changes in heart rate and blood pressure and feelings of fear, anxiety, or panic. Symptoms may come and go over short periods or last for a long time and can lead to fatigue and depression.

Exercises that involve moving the head and body in certain ways can help treat some balance disorders. Patient-specific exercises are developed by a physical therapist or other professional who understands balance and its relationship with other systems in the body.

Balance problems due to high blood pressure may be managed by eating less salt (sodium), maintaining a healthy weight , and exercising . Balance problems due to low blood pressure may be managed by drinking plenty of fluids such as water; avoiding alcohol ; and being cautious regarding your body’s posture and movement, such as never standing up too quickly. Consult with your doctor about making any changes in your diet or activity level.

Some people with a balance disorder may not be able to fully relieve their dizziness and will need to find ways to cope with it. A vestibular rehabilitation therapist can help develop an individualized treatment plan.

Chronic balance problems can affect all aspects of your life, including your relationships, work performance, and your ability to carry out daily activities. Support groups provide the opportunity to learn from other people with similar experiences and challenges.

If you have trouble with your balance, talk to your doctor about whether it’s safe to drive, and about ways to lower your risk of falling during daily activities, such as walking up or down stairs, using the bathroom, or exercising. To reduce your risk of injury from dizziness, do not walk in the dark. Avoid high heels and, instead, wear nonskid, rubber-soled, low-heeled shoes. Don’t walk on stairs or floors in socks or in shoes and slippers with smooth soles. If necessary, use a cane or walker. Make changes to add safety features at your home and workplace, such as adding handrails.

## You may also be interested in

• Learning more about falls and falls prevention
• Find out more about ways to prevent falls in certain rooms
• Watching a video on balance exercises

MedlinePlus National Library of Medicine       www.medlineplus.gov

Mayo Clinic www.mayoclinic.org/patient-care-and-health-information

National Institute on Deafness and Other Communication Disorders 800-241-1044 800-241-1055 (TTY) [email protected] www.nidcd.nih.gov

This content is provided by the NIH National Institute on Aging (NIA). NIA scientists and other experts review this content to ensure it is accurate and up to date.

Content reviewed: September 12, 2022

nia.nih.gov

An official website of the National Institutes of Health

#### IMAGES

1. Maths Problem Solving Strategies A4 Display Poster

2. Math Problem Solving Strategies

3. Math Problem Solving Strategies

4. Math Problem Solving Posters for Early Learners

5. Math Problem Solving Posters [Video]

6. What IS Problem-Solving?

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1. Problem solving

2. problem Solving Skills

3. Math Episode 18 (Problem Solving)

4. Problem Solving Strategies

5. Math Episode 17 (Creating Problems)

6. Problem solving

1. 6 Tips for Teaching Math Problem-Solving Skills

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 ...

2. PDF Teaching Strategies for Improving Algebra Knowledge in Middle and High

lgebra is often the first mathematics subject that requires extensive abstract thinking, a chal- lenging new skill for many students. Algebra moves students beyond an emphasis on arith-

3. Strengthening High School Students' Problem-Solving Skills

Finding, shaping, and solving problems puts high school students in charge of their learning and bolsters critical-thinking skills. As an educator for over 20 years, I've heard a lot about critical thinking, problem-solving, and inquiry and how they foster student engagement. However, I've also seen students draw a blank when they're ...

4. Math Problem Solving Strategies That Make Students Say "I Get It!"

Schema approach. This is a math intervention strategy that can make problem solving easier for all students, regardless of ability. Compare different word problems of the same type and construct a formula, or mathematical sentence stem, that applies to them all. For example, a simple subtraction problems could be expressed as:

5. Evidence-based math instruction: What you need to know

Math requires students to pay attention to details, plan, and self-monitor. Students also have to keep track of steps — and maybe even change direction while they work. Evidence-based math instruction helps these students because it breaks problems into multiple steps and reduces distractions.

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

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.

7. 4 Ways to Build Student-Centered Math Lessons

Encourage Productive Struggle. Problem-solving is an integral component of math, and allowing students to struggle productively as they attempt to solve complex problems "sends the message that the teacher believes students are capable of doing and creating mathematics," write Rhodes and Gareis. High school math teacher Solenne Abaziou, in ...

8. PDF Improving Math Performance (PDF)

procedural skills, and problem solving. Students are intellectually engaged in learning by reasoning, predicting, evaluating, concluding, and solving problems, skills that are fundamental for life-long learning. Other key strategies include: • A relentless focus on the California mathematics content standards.

9. Teaching Mathematics Through Problem Solving

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 ...

10. Strategies for Math Problem Solving

There are five strategies for math problem solving to word problems that you can teach your students in thirty minutes class. ... {English 5-9 and Mathematics 5-9}, and high school {Mathematics 6-12}. In 2013 I became a teacher author and started creating math curriculum for other teachers. I love teaching math to Elementary and Middle school ...

11. Problem Solving Activities: 7 Strategies

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!

12. Math Problem Solving Strategies

Math Problem Solving Strategies 1. C.U.B.E.S. C.U.B.E.S stands for circle the important numbers, underline the question, box the words that are keywords, eliminate extra information, and solve by showing work. ... especially in middle school and high school. Reading has to be done both at the literal and inferential levels to actually correctly ...

13. 9 Math Intervention Strategies for Struggling Students

Strategy 2: Use Schema-Based Instruction. Word problems can leave students across all grade levels wondering where to even start. The combination of parsing English and manipulating mathematical concepts can be daunting, especially for multilingual learners.One evidence-based strategy is to create a schema, or an underlying blueprint or structure that you can introduce students to and ...

14. Writing and choosing problems for a popular high school mathematics

In this paper, we consider the issues involved in creating appropriate problems for a popular mathematics competition, and how such problems differ from problems typically encountered in a classroom. We discuss the differences and similarities in school curricula versus the generally agreed upon topics encountered in international competitions. The question of inspiration for the development ...

15. Frontiers

Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional approaches supporting student learning in this area. This study aims to contribute to previous research by studying the effects of an instructional approach of cooperative learning on students' mathematical problem-solving in heterogeneous classrooms in ...

16. Problem Solving Strategies

Problem Solving Strategies - Math Challenge. If you are interested in adopting Math Challenge program for your school, please contact [email protected] and we'll be happy to assist you or someone you appointed to start on-boarding Math Challenge program into your school and provide all the necessary details.

17. 10 Strategies for Problem Solving in Math

Students may solve math issues in many ways, but here are ten math strategies for problem solving with high success rates. Depending on usage and preference, the strategies give kids renewed confidence as they work through difficulties. Understand the Problem. Before solving a math problem, kids need to know and understand their nature.

18. Methods Of Solving Problems In High School Mathematics

The authors consider all the most frequent methods of solving problems from the high-school course of mathematics. The fact that many problems are not followed by their solutions makes it possible to use the book for preparing for the entrance examinations to higher ... tsypkin-pinsky-methods-of-solving-problems-in-high-school-mathematics-mir ...

19. Module 1: Problem Solving Strategies

Step 2: Devise a plan. Going to use Guess and test along with making a tab. Many times the strategy below is used with guess and test. Make a table and look for a pattern: Procedure: Make a table reflecting the data in the problem.

20. Math Message Boards FAQ & Community Help

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21. Building Problem-Solving Skills Through 'Speed Dating'

In addition to hearing multiple ideas for solutions to their problems, kids get to know each other—and learn to see each other as experts in the room. "Problem speed dating ultimately teaches them that as a society or as a person, you're not alone," Beachboard says. "And reaching out to others is important because that's where the ...

22. Math Message Boards FAQ & Community Help

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23. Prentice Hall Retirement

Prentice Hall is part of Savvas Learning Company - a leading provider of K-12 education curriculum and digital learning solutions for students and educators. Due to Adobe's decision to stop supporting and updating Flash® in 2020, this site has been retired. We invite you to explore our featured secondary programs.

24. Problem Solving Strategies & Activities for Middle School Math

Much of math at the middle school level therefore takes place in the context of problem solving. As you move closer to algebra, geometry, and even trigonometry, your students will need to work ...

25. Math Message Boards FAQ & Community Help

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26. Older Adults and Balance Problems

Many older adults experience problems with balance and dizziness. Problems can be caused by certain medications, balance disorders, or other medical conditions. Balance problems are one reason older people fall. Maintaining good balance as you age and learning about fall prevention can help you get around, stay independent, and carry out daily ...

27. Math Message Boards FAQ & Community Help

Middle/High School Elementary School Online Books Recommendations Other Books & Gear All Products Gift Certificates. ... Small live classes for advanced math and language arts learners in grades 2-12. ... This can happen when you are logged in to Art of Problem Solving on multiple tabs of the same browser, and then sign out on one of those tabs