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Problem Solving Activities: 7 Strategies

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

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

I was so excited!

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

It was a proud moment for me!

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

Genius right? 

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

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

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

When I Finally Saw the Light

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

Problem Solving Activities

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

Seasonal Problem Solving

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

Cooperative Problem Solving Tasks

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

Notice and Wonder

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

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

Math Starters

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

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

Calculators

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

Three-Act Math Tasks

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

Getting the Most from Each of the Problem Solving Activities

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

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

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

Shametria Routt Banks

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2 Responses

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

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

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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

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

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

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

What are problem-solving strategies?

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

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

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

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

The ultimate guide to problem solving techniques

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

20 Math Strategies For Problem-Solving

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

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

Strategies to understand the problem

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

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

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

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

1. Read the problem aloud

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

2. Highlight keywords 

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

3. Summarize the information

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

4. Determine the unknown

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

5. Make a plan

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

Strategies for solving the problem 

1. draw a model or diagram.

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

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

2. Act it out

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

3. Work backwards

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

For example,

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

4. Write a number sentence

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

5. Use a formula

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

Strategies for checking the solution 

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

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

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

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

2. Estimate to check for reasonableness

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

3. Plug-In Method

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

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

4. Peer Review

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

5. Use a Calculator

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

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

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

Polya’s 4 steps include:

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

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

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

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

How Third Space Learning improves problem-solving 

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

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

One-on-one tutoring 

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

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

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

Problem-solving

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

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

READ MORE :

• 8 Common Core math examples
• Tier 3 Interventions: A School Leaders Guide
• Tier 2 Interventions: A School Leaders Guide
• Tier 1 Interventions: A School Leaders Guide

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

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

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

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

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Problem-Based Tasks in Math

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Providing students with opportunities to grapple with math has led to amazing things happening in my class. Students are totally excited and are driven to figure out not just how to solve a problem but why it works.

– Jessica Proffitt, Fifth-Grade Teacher at Two Rivers

Watch two rivers’s teachers and students at work on problem-based tasks in math.

Problem-Based Tasks Require Students to Apply Their Knowledge in New Contexts

Problem-based tasks are math lessons built around a single, compelling problem. The problems are truly “problematic” for students — that is, they do not offer an immediate solution.

The problems provide an opportunity for students to build conceptual understanding. Problem-based tasks require students to apply their current understanding and skills to new contexts that highlight core math concepts. For example, when students solve a problem that could be solved with multiplication before they have formally been taught what multiplication is and how it works, they build an understanding that multiplication is repeated addition.

Well-designed problem-based tasks provide multiple entry points for students to engage in problem solving, ensuring that all students have access to the same concepts. When students solve the problems in different ways—including drawing pictures, acting out the problem, writing algorithms, and using manipulatives—they make connections between the variety of models that all accurately illustrate the underlying mathematics.

Problem-Based Tasks in Math Resources

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

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

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

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

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

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

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

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

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

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

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

Key features of a good mathematics problem includes:

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

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

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

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

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

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

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

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

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

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

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

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

Number Talks

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

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

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

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

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

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

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

Instead, you and your students could say the following:

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

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

Using “Worksheets”

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

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

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

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

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

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

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

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

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

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

when we teach students how to problem solve

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

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

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

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

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• Authentic tasks
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Authentic tasks are designed to help students see mathematics as worthwhile and important. When students understand the purpose of a given problem in mathematics, they are more likely to persist when challenged. Authentic tasks generally have an ‘open middle’ which means that students can use different representations and solutions to communicate their knowledge and reasoning.

These curated links provide MAV members with access to nine authentic tasks from some of our primary consultants’ favourite resources. The 11 criteria provide MAV members with a research-informed context to consider each task’s potential impact on student thinking, ways of working, attitudes towards mathematics, their knowledge and understanding.

The following criteria was used to select the tasks based on their potential:

Used with permission © Martin Holt Educational Consultant 2017

If you would like to learn more about this approach to assessing or using tasks contact [email protected]

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Teachers: Resources for Middle Grades (6-8)

The North Carolina Collaborative for Mathematics Learning (NC 2 ML) aims to support NC math educators in implementing the revised mathematics content standards in ways that align with what we know from research on students’ mathematical thinking, mathematics teaching, and teacher learning. To do so, we bring together mathematics educators to co-design research-based resources and professional learning opportunities.

6-8 Resources Home

First Week Problem Solving Tasks

The Instructional Frameworks at each grade level recommend spending the first week of school doing general, high cognitive demand tasks with students in order to establish strong communication practices (SMP 3). Students can be enculturated into the discourse, listening and writing practices essential for strong mathematical reasoning while working these problems.

Additional Supporting Articles

Herbel-Eisenmenn, B. & Breyfogle, M. (2005). Questioning our patterns of questioning. Mathematics Teaching in the Middle School, 10(9), 484-489.

Stephan, M. (2014). Establishing standards for mathematical practice. Mathematics Teaching in the Middle School, 19(9), 532-538.

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K-5 Math Centers

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

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

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

Open-ended math problem solving tasks:

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

What Makes These So Great?

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

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

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

How and When Can I Use Them?

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

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

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

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

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

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

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

• Read more about: K-5 Math Ideas

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3 Problem-Solving Math Activities

Scottie Altland · September 5, 2018 · 1 Comment

A problem is simply a “problem” because there is no immediate, known solution. Problem solving activities in mathematics extend well beyond traditional word problems .

You can provide your student with activities that promote application of math skills while “busting boredom” at the same time! Puzzles and riddles, patterns, and logic problems can all be valuable exercises for students at all levels of mathematics. By engaging in short, fun activities like these, you can help your student become a more skillful, resilient, and successful problem-solver.

When practicing problem-solving skills, be certain to give your student time to explore a problem on her own to see how they might get started. Then discuss their approach together. It is important to provide support during the problem-solving process by showing that you value their ideas and helping them to see that mistakes can be useful. You can do this by asking open-ended questions to help your student gain a starting point, focus on a particular strategy, or help see a pattern or relationship. Questions such as, “What have you done before like this?”, “What can be made from …?” or “What might happen if you change…?” may serve as prompts when they needs inspiration.

Try the activities below to boost your student’s problem-solving skills.

Download the activities here .

1) Toothpick Puzzles

Toothpick puzzles (also referred to as matchstick puzzles) provide students a visualization challenge by applying their knowledge of basic geometric shapes and orientations. The only supplies you need are a box of toothpicks, a workspace, and a puzzle to solve. The goal is for students to transform given geometric figures into others by adding, moving, or removing toothpicks. These puzzles range in complexity and can be found online or in math puzzle books. As an extension, challenge your student to create their own puzzle for someone else to solve.

Sample toothpick puzzles of varying difficulty:

Download solutions to this activity here.

2) Fencing Numbers

The goal of this activity is to create a border or “fence” around each numeral by connecting dots horizontally and vertically so that each digit is bordered by the correct number of line segments.

Print a sheet of dot paper .

Use pencils and scissors to cut the size grid you want to use.

This game can be modified for abilities by adjusting the size of the grid and amount of numerals written. For example, a beginning student might begin with a grid that is 5 x 5 dots with a total of four numerals, while a more advanced student might increase the grid to 7 x 7 dots with six to eight numerals.

Begin by writing the digits 0, 1, 2, and 3 spread repeatedly in between “squares” on the dot paper. Each digit represents the number of line segments that will surround that square. For instance, a square that contains a 3 would have line segments on three sides, and a square that contains a 2 would have line segments on two sides, and so on. See the example boards and solutions for a 5 x 5 grid below.

Beware; there may be multiple solutions for the same problem! Thus, encourage your student to replicate the same problem grid multiple times and look for different solutions. A more advanced student can be challenged to create their own problem. Can they make a grid with only one solution? Is it possible to make a problem with four or more possible solutions?

3) It’s Knot a Problem!

Exercise lateral thinking skills– solving a problem through an indirect and creative approach that is not immediately obvious. You need two people, two pieces of string (or yarn) about one meter long each (or long enough so the person who will wear it can easily step over it), and some empty space to move around. If possible, use two different colored pieces of string. Each person needs a piece of string with a loop tied in both ends so it can be worn like “handcuffs”. Before tying off the loop on the second wrist, the participants loop the string around each other so they are hooked together. The figure below illustrates how the strings should appear when completed.

The goal is to unhook the strings while following these guidelines:

1) The string must remain tied and may not be removed from either participant’s wrists. 2) The string cannot be broken, cut, or damaged in any way.

Caution! This activity not only tests problem-solving skills, but it also promotes positive communication, teamwork, and persistence.

Problem-solving skills are not always taught directly but often learned indirectly through experience and practice. When incorporating problem solving activities aim to make them open-ended and playful to keep your student engaged. Incorporating fun activities like these from time to time foster creative and flexible thinking and can help your student transfer problem solving skills to other subject areas. By providing guidance and helping your student to see a problem from different perspectives, you will help foster a positive disposition towards problem-solving. As your student continues to learn how to effectively solve problems, they increase their understanding of the world around them and develop the tools they need to make decisions about the way they approach a problem.

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February 25, 2020 at 11:13 am

The ideas are very brilliant it encourages critical thinking and also help student think for a solution. Awesome!😍

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9 Fun And Engaging Math Problem Solving Activities Your Students Will Enjoy

Are you looking for math problem solving activities that are fun and engaging? Then continue reading on! I will be sharing with you 9 fun math problem solving activities that you can use in your class.  

What are mathematics problem-solving activities?

According to the National Council Of Teachers Of Mathematics, Mathematics problem solving refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. 

Problem-solving is a skill that we try to teach to our students in math class. A lot of times we will use word problems as problem-solving tasks. But there are actually more activities that do not involve story problems. 

You can use these problem-solving activities as a lesson themselves, math starters, review, fast finishers, with small groups or a large group.  

9 Fun Math Problem Solving Activities

Students often dread doing math word problems and tasks that are challenging. And forcing them down their throat is not the long-term solution as it can lead to math anxiety.

There must be a better way!

And the solution is…to find a fun way to tackle them!

Here is a list of 9 different ways to do problem-solving tasks. And I even gave some educational materials that you can grab if you are interested to use them in your class.

• Online Word Problems Practice
• Short Video
• Non-Routine Word Problems
• Hands-On Math Problem Solving Activities
• Math Puzzles
• Mystery Puzzles
• Scavenger Hunt
• Digital Treasure Hunt
• Escape Room

1) Online Word Problems Practice

Children love to go online. So by giving them a chance to play with the tablet or computer, they will already be more interested in the task on hand than usual. 

Consider the digital interactive task cards available on the Boom Learning site. They are often self-checking and require no preparation. This means they do not require much time from you and students can accomplish the mathematical practice independently.  

Furthermore, if you assign the Boom Cards to students, you can look through the reports of your student’s progress and results.   

These digital versions of word problems not only add a bit more fun to them but also help to develop a deeper understanding of mathematical concepts.

2) Short Video

Video provides a multisensory experience that helps to capture students’ attention. It is also great for memory retention and can enhance their learning experience. 

A) Show short videos that help them build their problem-solving skills. 

For example, matchstick puzzle examples. 

Related read: 3 Free Math Puzzles With Answer For You To Enjoy This Summer

B) Show them videos that teach them math skills or review math skills. 

This can be just a short review or a math hook for more math practice. 

Related read: 5 Hooks For Math Lessons That Will Engage Your Students Easily & Quickly

C) Show them a real-life problem and ask them to solve it using math.

Linking math to a real-life issue can always help to make math lessons more exciting. 

You can show them an existing issue and let them brainstorm on how to solve them. How can we use our math knowledge or other knowledge to solve it? (Sounds familiar? Consider project-based learning.)

Or you can show how real-life problems were solved due to our knowledge of math. Will they be the next mathematicians that make an impact on the world? 

3) Non-Routine Word Problems

What is more challenging and interesting than word problems? It’s non-routine word problems! 

They can be tricky and require different problem-solving strategies than the usual problem-solving approach. 

It requires some critical thinking to get to the correct answer. Sometimes there may also be different solutions to these challenging problems.

4) Hands-On Math Problem Solving Activities

By incorporating hands-on activities with word problems, word problems look more attractive now! 

Furthermore, kinesthetic learners will benefit greatly from math craft or math craftivity. Hands-on activities are engaging. 

Be aware of the suitability of the craft as young children or older students may require different sets of activities. One way to differentiate is by grade level. 

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5) Math Puzzles

There are many types of math puzzles. For example, logic puzzles, sudoku puzzles, and magic squares.

These math puzzles can help build logical reasoning. 

6) Mystery Puzzles

Students get to practice rigorous word problems and develop a deep conceptual understanding with these mystery puzzles!  

Students now have to solve word problems to know which are the correct clues. 

Furthermore, these worksheets are differentiated which means students of different standards can also utilize them. There are different culprits for the different sets which means students can do all of them if needed. 

7) Scavenger Hunt

Scavenger hunts are great movement activities for students. However, to incorporate word problems with a scavenger hunt, I would prefer to use them for lower elementary students. 

That’s because word problems for lower grades are usually shorter and require less time to solve. 

After all, if students have to stand for very long at a spot, it lowers the fun factors of the scavenger hunts.   

8) Digital Treasure Hunt

Treasure hunt is similar to a scavenger hunt. But what I have in mind for you is a digital treasure hunt that requires students to solve word problems prior to “digging” the spot. 

These digital versions of treasure hunting help you save some hassle but still engage students. 

9) Escape Room

Escape room is great for practicing problem solving skills as it usually includes a variety of problems and puzzles. The types of problems will vary, depending on the creator. So choose the ones that suit your students’ needs. 

Some elaborate escape rooms let students practice decision-making skills, collaboration skills, spatial reasoning, logical reasoning, deductive reasoning, and/or a variety of mathematical knowledge. 

Of course, we can always stick to the less fussy way and make students solve logic problems.

Final Thoughts

To make math problem-solving activity fun and engaging, the questions must be either interesting enough or within the student’s ability. 

The fun part of any puzzle is always those that we can solve if we think harder or out of the box. 

If it is too hard, students will get discouraged very soon and all of us will not meet our goals.

However, we also need to develop students’ growth mindset so that even if they can’t solve complex tasks, they will have the correct mindset facing their “failure”.

Hopefully, by using these ideas and tips mentioned above, your class will start looking forward to problem-solving activities. And we can also start looking forward to an increase in their math abilities and test scores! 

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Inside Problem Solving

The Inside Problem Solving problems are non-routine math problems designed to promote problem-solving in your classroom. Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity. The problems were developed by the Silicon Valley Mathematics Initiative and are aligned to the Common Core standards.

To request the Inside Problem Solving Solutions Guide, please get in touch with us via the feedback form .

Courtney’s Collection   Cut It Out Cutting a Cube Digging Dinosaurs Diminishing Return First Rate Friends You Can Count On Game Show Got Your Number Growing Staircases Measuring Mammals Measuring Up Miles of Tiles Movin ‘n Groovin On Balance Once Upon A Time Part and Whole Party Time Piece it Together Polly Gone Rod Trains Surrounded and Covered Squirreling It Away The Shape of Things The Wheel Shop Through the Grapevine Tri-Triangles What’s Your Angle?

Cutting a Cube (K.G.B.4) Digging Dinosaurs (K.OA.A.2) First Rate (K.CC.B.5, K.CC.C.6) Growing Staircases (K.CC.B.5) On Balance (K.MD.A.2)

Cutting a Cube (1.G.A.1) Growing Staircases (1.OA.A.1) Rod Trains (1.MD.A.2, 1.OA.C.6) Measuring Mammals (1.MD.A.1) Miles of Tiles (1.OA.A.1) Movin ‘n Groovin (1.OA.A.1) Piece it Together (1.G.A.2)

Courtney’s Collection (2.MD.C.8) Digging Dinosaurs (2.MD.C.8) Got Your Number (2.OA.B.2, 2.NBT.A.1, 2.NBT.A.4, 2.NBT.B.5) Miles of Tiles (2.NBT.B.5) Part and Whole (2.G.A.3) Piece it Together (2.G.A.1) Squirreling It Away (2.OA.1) The Shape of Things (2.G.A.1) Through the Grapevine (2.MD.D.9, 2.MD.D.10) What’s Your Angle? (2.G.A.1)

Measuring Up (3.OA.A.3) Once Upon A Time (3.MD.A.1) Part and Whole (3.G.A.2, 3.NF.A.1, 3.MD.C.6) Party Time (3.OA.A.3) Piece it Together (3.MD.C.5, 3.MD.D.8) Polly Gone (3.MD.D.8) Surrounded and Covered (3.MD.C.6, 3.MD.D.8) The Wheel Shop (3.OA.A.1, 3.OA.A.2) Tri-Triangles (3.OA.A.3)

Courtney’s Collection (4.MD.A.2) Digging Dinosaurs (4.MD.A.2) Diminishing Return (4.OA.A.3, 4.MD.A.2) Friends You Can Count On (4.OA.A.3) Game Show (4.OA.C.5) Growing Staircases (4.OA.C.5) Measuring Mammals (4.OA.A.2) Measuring Up (4.OA.A.3) Once Upon A Time (4.OA.A.3) Part and Whole (4.G.A.3) Party Time (4.NF.B.4c) Piece it Together (4.G.A.2, 4.MD.C.6) Squirreling It Away (4.OA.3) The Shape of Things (4.G.A.3) The Wheel Shop (4.OA.A.3) Tri-Triangles (4.OA.C.5)

Digging Dinosaurs (5.NBT.B.7) Movin ‘n Groovin (5.NF.B.4)

Courtney’s Collection (6.NS.B.4) Cutting a Cube (6.G.A.4, 6.RP.A.3c) Diminishing Return (6.RP.A.3a, 6.RP.A.3b) First Rate (6.RP.A.3b, 6.RP.A.2) Measuring Up (6.RP.A.3c, 6.EE.A.1, 6.EE.B.7) On Balance (6.EE.B.5, 6.EE.B.6, 6.EE.B.8) Once Upon A Time (6.NS.B.2, 6.NS.B.4) Movin ‘n Groovin (6.RP.A.3d) Part and Whole (6.G.A.1) Piece it Together (6.G.A.4) Polly Gone (6.G.A.1) Surrounded and Covered (6.RP.A.2, 6.RP.A.3b) Tri-Triangles (6.EE.A.1, 6.EE.B.6, 6.EE.C.9)

Courtney’s Collection (7.SP.C.8b) First Rate (7.RP.A.2b, 7.RP.A.3, 7.EE.B.4a) Friends You Can Count On (7.SP.C.7a, 7.SP.C.8a, 7.SP.C.8b) Game Show (7.SP.C.8a, 7.SP.C.8b) Got Your Number (7.NS.A.3) Measuring Mammals (7.RP.A.2a, 7.RP.A.2b, 7.RP.A.2c 7.RP.A.1) Measuring Up (7.RP.A.2b, 7.RP.A.2c, 7.RP.A.3, 7.EE.B.4) Movin ‘n Groovin (7.RP.A.2c, 7.RP.A.3) Part and Whole (7.NS.A.1D) Piece it Together (7.G.B.6) Polly Gone (7.G.B.6, 7.G.B.4) Rod Trains (7.SP.C.8b) Squirreling It Away (7.SP.8b) Surrounded and Covered (7.G.B.4, 7.G.B.6) Through the Grapevine (7.SP.A.2)

Cutting a Cube (8.G.A.1a) Digging Dinosaurs (8.EE.C.7b, 8.F.B.4) Diminishing Return (8.EE.C.7.b) Miles of Tiles (8.EE.C.8b, 8.EE.C.8c) Movin ‘n Groovin (8.EE.B5) On Balance (8.EE.C.8b, 8.EE.C.8c) Once Upon A Time (8.EE.C.8b) Squirreling It Away (8-F.1) Through the Grapevine (8.SP.A.1, 8.SP.A.2) The Wheel Shop (8.EE.C.8b, 8.EE.C.8c)

Courtney’s Collection (A-CED.A.2) Digging Dinosaurs (A-CED.A.2) Diminishing Return (A-CED.A.1) Growing Staircases (A-CED.A.2) Measuring Mammals (A-CED.A.2, A-REI.B.3, A-REI.C.6) Measuring Up (A-CED.2) Miles of Tiles (A-APR.A.1, A-SSE.A.1a, A-SSE.A.2) On Balance (A-CED.A.2, A-REI.C.6) Once Upon A Time (A-CED.A.1) Part and Whole (A-APR.D.6) Polly Gone (A-REI.C.6) Squirreling It Away (A-CED.2, A-CED.3, A-REI.6, A-REI.8, A-REI.10) The Wheel Shop (A-REI.C.6, A-REI.D.12) Tri-Triangles (A-CED.A.1, A-REI.B.4b, A-SSE.A.2)

Cut It Out (F-BF.A.1a) Digging Dinosaurs (F-IF.C.7b, F-IF.C.7e) Diminishing Return (F-BF.A.1a) First Rate (F-IF.B.6, F-BF.A.1a) Growing Staircases (F-LE.A.2, F-BF.A.2, F-BF.A.1a) Movin ‘n Groovin (F.BF. A.1a) Rod Trains (F-BF.A.1a) Squirreling It Away (F.LE.2, F-BF.1a, F-BF.2) Surrounded and Covered (F-BF.A.1a) Tri-Triangles (F-BF.A.1a) What’s Your Angle? (F-BF.A.1a)

Cut It Out (G-CO.B.6) Growing Staircases (G-MG.1) First Rate (G-SRT.C.8) Measuring Mammals (G-SRT.B.5) Miles of Tiles (G-MG.A.3) Once Upon A Time (G-C.A.2) Piece it Together (G.MG.A.1, G-MG.A.3, G.GMD.A.1, G.SRT.C.8) Polly Gone (G-CO.B.7, G-GPE.B.7, G-MG.A.3, G-GPE.B.4) The Shape of Things (G-C.A.2, G-CO.C.10, G-CO.C.11, G-SRT.B.5, G-MG.A.1) What’s Your Angle? (G-MG.A.3, G-C.A.2)

Digging Dinosaurs (S-ID.6.a) Diminishing Return (S-CP.A.2, S-CP.B.8) Friends You Can Count On (S-CP.A.4, S-CP.A.5, S-CP.B.6) Game Show (S-MD.A.1, S-MD.A.2, S-MD.A.3) Growing Staircases (S-ID.6a) Party Time (S-CP.A.1, S-CP.B.9, S-CP.B.8) Squirreling It Away (S-ID.6a) Through the Grapevine (S-IC.B.4, S-ID.A.1, S-ID.A.2, S-ID.A.3, S-ID.B.5, S-ID.B.6c) The Wheel Shop (S-CP.A.1)

Why Problem Solving?

Problem solving is the cornerstone of doing mathematics. George Polya, a famous mathematician from Stanford, once said, "A problem is not a problem if you can solve it in 24 hours." His point was that a problem that you can solve in less than a day is usually a problem that is similar to one that you have solved before, or at least is one where you recognize that a certain approach would lead to the solution. Bu t in real life, a problem is a situation that confronts you and you don’t have an idea of where to even start. Mathematics is the toolbox that solves so many problems. Whether it is calculating an estimate measure, modeling a complex situation, determining the probability of a chance event, transforming a graphical image or proving a case using deductive reasoning, mathematics is used. If we want our student s to be problem solvers and mathematically powerful, we must model perseverance and challenge students with non-routine problems.

Problem Solving in Mathematics Education

• Open Access
• First Online: 28 June 2016

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• Peter Liljedahl 6 ,
• Manuel Santos-Trigo 7 ,
• Uldarico Malaspina 8 &
• Regina Bruder 9  

Part of the book series: ICME-13 Topical Surveys ((ICME13TS))

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Problem solving in mathematics education has been a prominent research field that aims at understanding and relating the processes involved in solving problems to students’ development of mathematical knowledge and problem solving competencies. The accumulated knowledge and field developments include conceptual frameworks to characterize learners’ success in problem solving activities, cognitive, metacognitive, social and affective analysis, curriculum proposals, and ways to foster problem solving approaches. In the survey, four interrelated areas are reviewed: (i) the relevance of heuristics in problem solving approaches—why are they important and what research tells us about their use? (ii) the need to characterize and foster creative problem solving approaches—what type of heuristics helps learners think of and practice creative solutions? (iii) the importance for learners to formulate and pursue their own problems; and (iv) the role played by the use of both multiple purpose and ad hoc mathematical action types of technologies in problem solving activities—what ways of reasoning do learners construct when they rely on the use of digital technologies and how technology and technology approaches can be reconciled?

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• Mathematical Problem
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• Creative Process
• Digital Technology
• Mathematical Task

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Mathematical problem solving has long been seen as an important aspect of mathematics, the teaching of mathematics, and the learning of mathematics. It has infused mathematics curricula around the world with calls for the teaching of problem solving as well as the teaching of mathematics through problem solving. And as such, it has been of interest to mathematics education researchers for as long as our field has existed. More relevant, mathematical problem solving has played a part in every ICME conference, from 1969 until the forthcoming meeting in Hamburg, wherein mathematical problem solving will reside most centrally within the work of Topic Study 19: Problem Solving in Mathematics Education. This booklet is being published on the occasion of this Topic Study Group.

To this end, we have assembled four summaries looking at four distinct, yet inter-related, dimensions of mathematical problem solving. The first summary, by Regina Bruder, is a nuanced look at heuristics for problem solving. This notion of heuristics is carried into Peter Liljedahl’s summary, which looks specifically at a progression of heuristics leading towards more and more creative aspects of problem solving. This is followed by Luz Manuel Santos Trigo’s summary introducing us to problem solving in and with digital technologies. The last summary, by Uldarico Malaspina Jurado, documents the rise of problem posing within the field of mathematics education in general and the problem solving literature in particular.

Each of these summaries references in some critical and central fashion the works of George Pólya or Alan Schoenfeld. To the initiated researchers, this is no surprise. The seminal work of these researchers lie at the roots of mathematical problem solving. What is interesting, though, is the diverse ways in which each of the four aforementioned contributions draw on, and position, these works so as to fit into the larger scheme of their respective summaries. This speaks to not only the depth and breadth of these influential works, but also the diversity with which they can be interpreted and utilized in extending our thinking about problem solving.

Taken together, what follows is a topical survey of ideas representing the diversity of views and tensions inherent in a field of research that is both a means to an end and an end onto itself and is unanimously seen as central to the activities of mathematics.

1 Survey on the State-of-the-Art

1.1 role of heuristics for problem solving—regina bruder.

The origin of the word heuristic dates back to the time of Archimedes and is said to have come out of one of the famous stories told about this great mathematician and inventor. The King of Syracuse asked Archimedes to check whether his new wreath was really made of pure gold. Archimedes struggled with this task and it was not until he was at the bathhouse that he came up with the solution. As he entered the tub he noticed that he had displaced a certain amount of water. Brilliant as he was, he transferred this insight to the issue with the wreath and knew he had solved the problem. According to the legend, he jumped out of the tub and ran from the bathhouse naked screaming, “Eureka, eureka!”. Eureka and heuristic have the same root in the ancient Greek language and so it has been claimed that this is how the academic discipline of “heuristics” dealing with effective approaches to problem solving (so-called heurisms) was given its name. Pólya ( 1964 ) describes this discipline as follows:

Heuristics deals with solving tasks. Its specific goals include highlighting in general terms the reasons for selecting those moments in a problem the examination of which could help us find a solution. (p. 5)

This discipline has grown, in part, from examining the approaches to certain problems more in detail and comparing them with each other in order to abstract similarities in approach, or so-called heurisms. Pólya ( 1949 ), but also, inter alia, Engel ( 1998 ), König ( 1984 ) and Sewerin ( 1979 ) have formulated such heurisms for mathematical problem tasks. The problem tasks examined by the authors mentioned are predominantly found in the area of talent programmes, that is, they often go back to mathematics competitions.

In 1983 Zimmermann provided an overview of heuristic approaches and tools in American literature which also offered suggestions for mathematics classes. In the German-speaking countries, an approach has established itself, going back to Sewerin ( 1979 ) and König ( 1984 ), which divides school-relevant heuristic procedures into heuristic tools, strategies and principles, see also Bruder and Collet ( 2011 ).

Below is a review of the conceptual background of heuristics, followed by a description of the effect mechanisms of heurisms in problem-solving processes.

1.1.1 Research Review on the Promotion of Problem Solving

In the 20th century, there has been an advancement of research on mathematical problem solving and findings about possibilities to promote problem solving with varying priorities (c.f. Pehkonen 1991 ). Based on a model by Pólya ( 1949 ), in a first phase of research on problem solving, particularly in the 1960s and the 1970s, a series of studies on problem-solving processes placing emphasis on the importance of heuristic strategies (heurisms) in problem solving has been carried out. It was assumed that teaching and learning heuristic strategies, principles and tools would provide students with an orientation in problem situations and that this could thus improve students’ problem-solving abilities (c.f. for instance, Schoenfeld 1979 ). This approach, mostly researched within the scope of talent programmes for problem solving, was rather successful (c.f. for instance, Sewerin 1979 ). In the 1980s, requests for promotional opportunities in everyday teaching were given more and more consideration: “ problem solving must be the focus of school mathematics in the 1980s ” (NCTM 1980 ). For the teaching and learning of problem solving in regular mathematics classes, the current view according to which cognitive, heuristic aspects were paramount, was expanded by certain student-specific aspects, such as attitudes, emotions and self-regulated behaviour (c.f. Kretschmer 1983 ; Schoenfeld 1985 , 1987 , 1992 ). Kilpatrick ( 1985 ) divided the promotional approaches described in the literature into five methods which can also be combined with each other.

Osmosis : action-oriented and implicit imparting of problem-solving techniques in a beneficial learning environment

Memorisation : formation of special techniques for particular types of problem and of the relevant questioning when problem solving

Imitation : acquisition of problem-solving abilities through imitation of an expert

Cooperation : cooperative learning of problem-solving abilities in small groups

Reflection : problem-solving abilities are acquired in an action-oriented manner and through reflection on approaches to problem solving.

Kilpatrick ( 1985 ) views as success when heuristic approaches are explained to students, clarified by means of examples and trained through the presentation of problems. The need of making students aware of heuristic approaches is by now largely accepted in didactic discussions. Differences in varying approaches to promoting problem-solving abilities rather refer to deciding which problem-solving strategies or heuristics are to imparted to students and in which way, and not whether these should be imparted at all or not.

1.1.2 Heurisms as an Expression of Mental Agility

The activity theory, particularly in its advancement by Lompscher ( 1975 , 1985 ), offers a well-suited and manageable model to describe learning activities and differences between learners with regard to processes and outcomes in problem solving (c.f. Perels et al. 2005 ). Mental activity starts with a goal and the motive of a person to perform such activity. Lompscher divides actual mental activity into content and process. Whilst the content in mathematical problem-solving consists of certain concepts, connections and procedures, the process describes the psychological processes that occur when solving a problem. This course of action is described in Lompscher by various qualities, such as systematic planning, independence, accuracy, activity and agility. Along with differences in motivation and the availability of expertise, it appears that intuitive problem solvers possess a particularly high mental agility, at least with regard to certain contents areas.

According to Lompscher, “flexibility of thought” expresses itself

… by the capacity to change more or less easily from one aspect of viewing to another one or to embed one circumstance or component into different correlations, to understand the relativity of circumstances and statements. It allows to reverse relations, to more or less easily or quickly attune to new conditions of mental activity or to simultaneously mind several objects or aspects of a given activity (Lompscher 1975 , p. 36).

These typical manifestations of mental agility can be focused on in problem solving by mathematical means and can be related to the heurisms known from the analyses of approaches by Pólya et al. (c.f. also Bruder 2000 ):

Reduction : Successful problem solvers will intuitively reduce a problem to its essentials in a sensible manner. To achieve such abstraction, they often use visualisation and structuring aids, such as informative figures, tables, solution graphs or even terms. These heuristic tools are also very well suited to document in retrospect the approach adopted by the intuitive problem solvers in a way that is comprehensible for all.

Reversibility : Successful problem solvers are able to reverse trains of thought or reproduce these in reverse. They will do this in appropriate situations automatically, for instance, when looking for a key they have mislaid. A corresponding general heuristic strategy is working in reverse.

Minding of aspects : Successful problem solvers will mind several aspects of a given problem at the same time or easily recognise any dependence on things and vary them in a targeted manner. Sometimes, this is also a matter of removing barriers in favour of an idea that appears to be sustainable, that is, by simply “hanging on” to a certain train of thought even against resistance. Corresponding heurisms are, for instance, the principle of invariance, the principle of symmetry (Engel 1998 ), the breaking down or complementing of geometric figures to calculate surface areas, or certain terms used in binomial formulas.

Change of aspects : Successful problem solvers will possibly change their assumptions, criteria or aspects minded in order to find a solution. Various aspects of a given problem will be considered intuitively or the problem be viewed from a different perspective, which will prevent “getting stuck” and allow for new insights and approaches. For instance, many elementary geometric propositions can also be proved in an elegant vectorial manner.

Transferring : Successful problem solvers will be able more easily than others to transfer a well-known procedure to another, sometimes even very different context. They recognise more easily the “framework” or pattern of a given task. Here, this is about own constructions of analogies and continual tracing back from the unknown to the known.

Intuitive, that is, untrained good problem solvers, are, however, often unable to access these flexibility qualities consciously. This is why they are also often unable to explain how they actually solved a given problem.

To be able to solve problems successfully, a certain mental agility is thus required. If this is less well pronounced in a certain area, learning how to solve problems means compensating by acquiring heurisms. In this case, insufficient mental agility is partly “offset” through the application of knowledge acquired by means of heurisms. Mathematical problem-solving competences are thus acquired through the promotion of manifestations of mental agility (reduction, reversibility, minding of aspects and change of aspects). This can be achieved by designing sub-actions of problem solving in connection with a (temporarily) conscious application of suitable heurisms. Empirical evidence for the success of the active principle of heurisms has been provided by Collet ( 2009 ).

Against such background, learning how to solve problems can be established as a long-term teaching and learning process which basically encompasses four phases (Bruder and Collet 2011 ):

Intuitive familiarisation with heuristic methods and techniques.

Making aware of special heurisms by means of prominent examples (explicit strategy acquisition).

Short conscious practice phase to use the newly acquired heurisms with differentiated task difficulties.

Expanding the context of the strategies applied.

In the first phase, students are familiarised with heurisms intuitively by means of targeted aid impulses and questions (what helped us solve this problem?) which in the following phase are substantiated on the basis of model tasks, are given names and are thus made aware of their existence. The third phase serves the purpose of a certain familiarisation with the new heurisms and the experience of competence through individualised practising at different requirement levels, including in the form of homework over longer periods. A fourth and delayed fourth phase aims at more flexibility through the transfer to other contents and contexts and the increasingly intuitive use of the newly acquired heurisms, so that students can enrich their own problem-solving models in a gradual manner. The second and third phases build upon each other in close chronological order, whilst the first phase should be used in class at all times.

All heurisms can basically be described in an action-oriented manner by means of asking the right questions. The way of asking questions can thus also establish a certain kind of personal relation. Even if the teacher presents and suggests the line of basic questions with a prototypical wording each time, students should always be given the opportunity to find “their” wording for the respective heurism and take a note of it for themselves. A possible key question for the use of a heuristic tool would be: How to illustrate and structure the problem or how to present it in a different way?

Unfortunately, for many students, applying heuristic approaches to problem solving will not ensue automatically but will require appropriate early and long-term promoting. The results of current studies, where promotion approaches to problem solving are connected with self-regulation and metacognitive aspects, demonstrate certain positive effects of such combination on students. This field of research includes, for instance, studies by Lester et al. ( 1989 ), Verschaffel et al. ( 1999 ), the studies on teaching method IMPROVE by Mevarech and Kramarski ( 1997 , 2003 ) and also the evaluation of a teaching concept on learning how to solve problems by the gradual conscious acquisition of heurisms by Collet and Bruder ( 2008 ).

1.2 Creative Problem Solving—Peter Liljedahl

There is a tension between the aforementioned story of Archimedes and the heuristics presented in the previous section. Archimedes, when submersing himself in the tub and suddenly seeing the solution to his problem, wasn’t relying on osmosis, memorisation, imitation, cooperation, or reflection (Kilpatrick 1985 ). He wasn’t drawing on reduction, reversibility, minding of aspects, change of aspect, or transfer (Bruder 2000 ). Archimedes was stuck and it was only, in fact, through insight and sudden illumination that he managed to solve his problem. In short, Archimedes was faced with a problem that the aforementioned heuristics, and their kind, would not help him to solve.

According to some, such a scenario is the definition of a problem. For example, Resnick and Glaser ( 1976 ) define a problem as being something that you do not have the experience to solve. Mathematicians, in general, agree with this (Liljedahl 2008 ).

Any problem in which you can see how to attack it by deliberate effort, is a routine problem, and cannot be an important discover. You must try and fail by deliberate efforts, and then rely on a sudden inspiration or intuition or if you prefer to call it luck. (Dan Kleitman, participant cited in Liljedahl 2008 , p. 19).

Problems, then, are tasks that cannot be solved by direct effort and will require some creative insight to solve (Liljedahl 2008 ; Mason et al. 1982 ; Pólya 1965 ).

1.2.1 A History of Creativity in Mathematics Education

In 1902, the first half of what eventually came to be a 30 question survey was published in the pages of L’Enseignement Mathématique , the journal of the French Mathematical Society. The authors, Édouard Claparède and Théodore Flournoy, were two Swiss psychologists who were deeply interested in the topics of mathematical discovery, creativity and invention. Their hope was that a widespread appeal to mathematicians at large would incite enough responses for them to begin to formulate some theories about this topic. The first half of the survey centered on the reasons for becoming a mathematician (family history, educational influences, social environment, etc.), attitudes about everyday life, and hobbies. This was eventually followed, in 1904, by the publication of the second half of the survey pertaining, in particular, to mental images during periods of creative work. The responses were sorted according to nationality and published in 1908.

During this same period Henri Poincaré (1854–1912), one of the most noteworthy mathematicians of the time, had already laid much of the groundwork for his own pursuit of this same topic and in 1908 gave a presentation to the French Psychological Society in Paris entitled L’Invention mathématique —often mistranslated to Mathematical Creativity Footnote 1 (c.f. Poincaré 1952 ). At the time of the presentation Poincaré stated that he was aware of Claparède and Flournoy’s work, as well as their results, but expressed that they would only confirm his own findings. Poincaré’s presentation, as well as the essay it spawned, stands to this day as one of the most insightful, and thorough treatments of the topic of mathematical discovery, creativity, and invention.

Just at this time, I left Caen, where I was living, to go on a geological excursion under the auspices of the School of Mines. The incident of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuschian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had the time, as, upon taking my seat in the omnibus, I went on with the conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake, I verified the results at my leisure. (Poincaré 1952 , p. 53)

So powerful was his presentation, and so deep were his insights into his acts of invention and discovery that it could be said that he not so much described the characteristics of mathematical creativity, as defined them. From that point forth mathematical creativity, or even creativity in general, has not been discussed seriously without mention of Poincaré’s name.

Inspired by this presentation, Jacques Hadamard (1865–1963), a contemporary and a friend of Poincaré’s, began his own empirical investigation into this fascinating phenomenon. Hadamard had been critical of Claparède and Flournoy’s work in that they had not adequately treated the topic on two fronts. As exhaustive as the survey appeared to be, Hadamard felt that it failed to ask some key questions—the most important of which was with regard to the reason for failures in the creation of mathematics. This seemingly innocuous oversight, however, led directly to his second and “most important criticism” (Hadamard 1945 ). He felt that only “first-rate men would dare to speak of” (p. 10) such failures. So, inspired by his good friend Poincaré’s treatment of the subject Hadamard retooled the survey and gave it to friends of his for consideration—mathematicians such as Henri Poincaré and Albert Einstein, whose prominence were beyond reproach. Ironically, the new survey did not contain any questions that explicitly dealt with failure. In 1943 Hadamard gave a series of lectures on mathematical invention at the École Libre des Hautes Études in New York City. These talks were subsequently published as The Psychology of Mathematical Invention in the Mathematical Field (Hadameard 1945 ).

Hadamard’s classic work treats the subject of invention at the crossroads of mathematics and psychology. It provides not only an entertaining look at the eccentric nature of mathematicians and their rituals, but also outlines the beliefs of mid twentieth-century mathematicians about the means by which they arrive at new mathematics. It is an extensive exploration and extended argument for the existence of unconscious mental processes. In essence, Hadamard took the ideas that Poincaré had posed and, borrowing a conceptual framework for the characterization of the creative process from the Gestaltists of the time (Wallas 1926 ), turned them into a stage theory. This theory still stands as the most viable and reasonable description of the process of mathematical creativity.

1.2.2 Defining Mathematical Creativity

The phenomena of mathematical creativity, although marked by sudden illumination, actually consist of four separate stages stretched out over time, of which illumination is but one stage. These stages are initiation, incubation, illumination, and verification (Hadamard 1945 ). The first of these stages, the initiation phase, consists of deliberate and conscious work. This would constitute a person’s voluntary, and seemingly fruitless, engagement with a problem and be characterized by an attempt to solve the problem by trolling through a repertoire of past experiences. This is an important part of the inventive process because it creates the tension of unresolved effort that sets up the conditions necessary for the ensuing emotional release at the moment of illumination (Hadamard 1945 ; Poincaré 1952 ).

Following the initiation stage the solver, unable to come up with a solution stops working on the problem at a conscious level and begins to work on it at an unconscious level (Hadamard 1945 ; Poincaré 1952 ). This is referred to as the incubation stage of the inventive process and can last anywhere from several minutes to several years. After the period of incubation a rapid coming to mind of a solution, referred to as illumination , may occur. This is accompanied by a feeling of certainty and positive emotions (Poincaré 1952 ). Although the processes of incubation and illumination are shrouded behind the veil of the unconscious there are a number of things that can be deduced about them. First and foremost is the fact that unconscious work does, indeed, occur. Poincaré ( 1952 ), as well as Hadamard ( 1945 ), use the very real experience of illumination, a phenomenon that cannot be denied, as evidence of unconscious work, the fruits of which appear in the flash of illumination. No other theory seems viable in explaining the sudden appearance of solution during a walk, a shower, a conversation, upon waking, or at the instance of turning the conscious mind back to the problem after a period of rest (Poincaré 1952 ). Also deducible is that unconscious work is inextricably linked to the conscious and intentional effort that precedes it.

There is another remark to be made about the conditions of this unconscious work: it is possible, and of a certainty it is only fruitful, if it is on the one hand preceded and on the other hand followed by a period of conscious work. These sudden inspirations never happen except after some days of voluntary effort which has appeared absolutely fruitless and whence nothing good seems to have come … (Poincaré 1952 , p. 56)

Hence, the fruitless efforts of the initiation phase are only seemingly so. They not only set up the aforementioned tension responsible for the emotional release at the time of illumination, but also create the conditions necessary for the process to enter into the incubation phase.

Illumination is the manifestation of a bridging that occurs between the unconscious mind and the conscious mind (Poincaré 1952 ), a coming to (conscious) mind of an idea or solution. What brings the idea forward to consciousness is unclear, however. There are theories of the aesthetic qualities of the idea, effective surprise/shock of recognition, fluency of processing, or breaking functional fixedness. For reasons of brevity I will only expand on the first of these.

Poincaré proposed that ideas that were stimulated during initiation remained stimulated during incubation. However, freed from the constraints of conscious thought and deliberate calculation, these ideas would begin to come together in rapid and random unions so that “their mutual impacts may produce new combinations” (Poincaré 1952 ). These new combinations, or ideas, would then be evaluated for viability using an aesthetic sieve, which allows through to the conscious mind only the “right combinations” (Poincaré 1952 ). It is important to note, however, that good or aesthetic does not necessarily mean correct. Correctness is evaluated during the verification stage.

The purpose of verification is not only to check for correctness. It is also a method by which the solver re-engages with the problem at the level of details. That is, during the unconscious work the problem is engaged with at the level of ideas and concepts. During verification the solver can examine these ideas in closer details. Poincaré succinctly describes both of these purposes.

As for the calculations, themselves, they must be made in the second period of conscious work, that which follows the inspiration, that in which one verifies the results of this inspiration and deduces their consequences. (Poincaré 1952 , p. 62)

Aside from presenting this aforementioned theory on invention, Hadamard also engaged in a far-reaching discussion on a number of interesting, and sometimes quirky, aspects of invention and discovery that he had culled from the results of his empirical study, as well as from pertinent literature. This discussion was nicely summarized by Newman ( 2000 ) in his commentary on the elusiveness of invention.

The celebrated phrenologist Gall said mathematical ability showed itself in a bump on the head, the location of which he specified. The psychologist Souriau, we are told, maintained that invention occurs by “pure chance”, a valuable theory. It is often suggested that creative ideas are conjured up in “mathematical dreams”, but this attractive hypothesis has not been verified. Hadamard reports that mathematicians were asked whether “noises” or “meteorological circumstances” helped or hindered research [..] Claude Bernard, the great physiologist, said that in order to invent “one must think aside”. Hadamard says this is a profound insight; he also considers whether scientific invention may perhaps be improved by standing or sitting or by taking two baths in a row. Helmholtz and Poincaré worked sitting at a table; Hadamard’s practice is to pace the room (“Legs are the wheels of thought”, said Emile Angier); the chemist J. Teeple was the two-bath man. (p. 2039)

1.2.3 Discourses on Creativity

Creativity is a term that can be used both loosely and precisely. That is, while there exists a common usage of the term there also exists a tradition of academic discourse on the subject. A common usage of creative refers to a process or a person whose products are original, novel, unusual, or even abnormal (Csíkszentmihályi 1996 ). In such a usage, creativity is assessed on the basis of the external and observable products of the process, the process by which the product comes to be, or on the character traits of the person doing the ‘creating’. Each of these usages—product, process, person—is the roots of the discourses (Liljedahl and Allan 2014 ) that I summarize here, the first of which concerns products.

Consider a mother who states that her daughter is creative because she drew an original picture. The basis of such a statement can lie either in the fact that the picture is unlike any the mother has ever seen or unlike any her daughter has ever drawn before. This mother is assessing creativity on the basis of what her daughter has produced. However, the standards that form the basis of her assessment are neither consistent nor stringent. There does not exist a universal agreement as to what she is comparing the picture to (pictures by other children or other pictures by the same child). Likewise, there is no standard by which the actual quality of the picture is measured. The academic discourse that concerns assessment of products, on the other hand, is both consistent and stringent (Csíkszentmihályi 1996 ). This discourse concerns itself more with a fifth, and as yet unmentioned, stage of the creative process; elaboration . Elaboration is where inspiration becomes perspiration (Csíkszentmihályi 1996 ). It is the act of turning a good idea into a finished product, and the finished product is ultimately what determines the creativity of the process that spawned it—that is, it cannot be a creative process if nothing is created. In particular, this discourse demands that the product be assessed against other products within its field, by the members of that field, to determine if it is original AND useful (Csíkszentmihályi 1996 ; Bailin 1994 ). If it is, then the product is deemed to be creative. Note that such a use of assessment of end product pays very little attention to the actual process that brings this product forth.

The second discourse concerns the creative process. The literature pertaining to this can be separated into two categories—a prescriptive discussion of the creativity process and a descriptive discussion of the creativity process. Although both of these discussions have their roots in the four stages that Wallas ( 1926 ) proposed makes up the creative process, they make use of these stages in very different ways. The prescriptive discussion of the creative process is primarily focused on the first of the four stages, initiation , and is best summarized as a cause - and - effect discussion of creativity, where the thinking processes during the initiation stage are the cause and the creative outcome are the effects (Ghiselin 1952 ). Some of the literature claims that the seeds of creativity lie in being able to think about a problem or situation analogically. Other literature claims that utilizing specific thinking tools such as imagination, empathy, and embodiment will lead to creative products. In all of these cases, the underlying theory is that the eventual presentation of a creative idea will be precipitated by the conscious and deliberate efforts during the initiation stage. On the other hand, the literature pertaining to a descriptive discussion of the creative process is inclusive of all four stages (Kneller 1965 ; Koestler 1964 ). For example, Csíkszentmihályi ( 1996 ), in his work on flow attends to each of the stages, with much attention paid to the fluid area between conscious and unconscious work, or initiation and incubation. His claim is that the creative process is intimately connected to the enjoyment that exists during times of sincere and consuming engagement with a situation, the conditions of which he describes in great detail.

The third, and final, discourse on creativity pertains to the person. This discourse is space dominated by two distinct characteristics, habit and genius. Habit has to do with the personal habits as well as the habits of mind of people that have been deemed to be creative. However, creative people are most easily identified through their reputation for genius. Consequently, this discourse is often dominated by the analyses of the habits of geniuses as is seen in the work of Ghiselin ( 1952 ), Koestler ( 1964 ), and Kneller ( 1965 ) who draw on historical personalities such as Albert Einstein, Henri Poincaré, Vincent Van Gogh, D.H. Lawrence, Samuel Taylor Coleridge, Igor Stravinsky, and Wolfgang Amadeus Mozart to name a few. The result of this sort of treatment is that creative acts are viewed as rare mental feats, which are produced by extraordinary individuals who use extraordinary thought processes.

These different discourses on creativity can be summed up in a tension between absolutist and relativist perspectives on creativity (Liljedahl and Sriraman 2006 ). An absolutist perspective assumes that creative processes are the domain of genius and are present only as precursors to the creation of remarkably useful and universally novel products. The relativist perspective, on the other hand, allows for every individual to have moments of creativity that may, or may not, result in the creation of a product that may, or may not, be either useful or novel.

Between the work of a student who tries to solve a problem in geometry or algebra and a work of invention, one can say there is only a difference of degree. (Hadamard 1945 , p. 104).

Regardless of discourse, however, creativity is not “part of the theories of logical forms” (Dewey 1938 ). That is, creativity is not representative of the lock-step logic and deductive reasoning that mathematical problem solving is often presumed to embody (Bibby 2002 ; Burton 1999 ). Couple this with the aforementioned demanding constraints as to what constitutes a problem, where then does that leave problem solving heuristics? More specifically, are there creative problem solving heuristics that will allow us to resolve problems that require illumination to solve? The short answer to this question is yes—there does exist such problem solving heuristics. To understand these, however, we must first understand the routine problem solving heuristics they are built upon. In what follows, I walk through the work of key authors and researchers whose work offers us insights into progressively more creative problem solving heuristics for solving true problems.

1.2.4 Problem Solving by Design

In a general sense, design is defined as the algorithmic and deductive approach to solving a problem (Rusbult 2000 ). This process begins with a clearly defined goal or objective after which there is a great reliance on relevant past experience, referred to as repertoire (Bruner 1964 ; Schön 1987 ), to produce possible options that will lead towards a solution of the problem (Poincaré 1952 ). These options are then examined through a process of conscious evaluations (Dewey 1933 ) to determine their suitability for advancing the problem towards the final goal. In very simple terms, problem solving by design is the process of deducing the solution from that which is already known.

Mayer ( 1982 ), Schoenfeld ( 1982 ), and Silver ( 1982 ) state that prior knowledge is a key element in the problem solving process. Prior knowledge influences the problem solver’s understanding of the problem as well as the choice of strategies that will be called upon in trying to solve the problem. In fact, prior knowledge and prior experiences is all that a solver has to draw on when first attacking a problem. As a result, all problem solving heuristics incorporate this resource of past experiences and prior knowledge into their initial attack on a problem. Some heuristics refine these ideas, and some heuristics extend them (c.f. Kilpatrick 1985 ; Bruder 2000 ). Of the heuristics that refine, none is more influential than the one created by George Pólya (1887–1985).

1.2.5 George Pólya: How to Solve It

In his book How to Solve It (1949) Pólya lays out a problem solving heuristic that relies heavily on a repertoire of past experience. He summarizes the four-step process of his heuristic as follows:

Understanding the Problem

First. You have to understand the problem.

What is the unknown? What are the data? What is the condition?

Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?

Draw a figure. Introduce suitable notation.

Separate the various parts of the condition. Can you write them down?

Devising a Plan

Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.

Have you seen it before? Or have you seen the same problem in a slightly different form?

Do you know a related problem? Do you know a theorem that could be useful?

Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.

Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?

Could you restate the problem? Could you restate it still differently? Go back to definitions.

If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?

Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

Carrying Out the Plan

Third. Carry out your plan.

Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

Looking Back

Fourth. Examine the solution obtained.

Can you check the result? Can you check the argument?

Can you derive the solution differently? Can you see it at a glance?

Can you use the result, or the method, for some other problem?

The emphasis on auxiliary problems, related problems, and analogous problems that are, in themselves, also familiar problems is an explicit manifestation of relying on a repertoire of past experience. This use of familiar problems also requires an ability to deduce from these related problems a recognizable and relevant attribute that will transfer to the problem at hand. The mechanism that allows for this transfer of knowledge between analogous problems is known as analogical reasoning (English 1997 , 1998 ; Novick 1988 , 1990 , 1995 ; Novick and Holyoak 1991 ) and has been shown to be an effective, but not always accessible, thinking strategy.

Step four in Pólya’s heuristic, looking back, is also a manifestation of utilizing prior knowledge to solve problems, albeit an implicit one. Looking back makes connections “in memory to previously acquired knowledge [..] and further establishes knowledge in long-term memory that may be elaborated in later problem-solving encounters” (Silver 1982 , p. 20). That is, looking back is a forward-looking investment into future problem solving encounters, it sets up connections that may later be needed.

Pólya’s heuristic is a refinement on the principles of problem solving by design. It not only makes explicit the focus on past experiences and prior knowledge, but also presents these ideas in a very succinct, digestible, and teachable manner. This heuristic has become a popular, if not the most popular, mechanism by which problem solving is taught and learned.

1.2.6 Alan Schoenfeld: Mathematical Problem Solving

The work of Alan Schoenfeld is also a refinement on the principles of problem solving by design. However, unlike Pólya ( 1949 ) who refined these principles at a theoretical level, Schoenfeld has refined them at a practical and empirical level. In addition to studying taught problem solving strategies he has also managed to identify and classify a variety of strategies, mostly ineffectual, that students invoke naturally (Schoenfeld 1985 , 1992 ). In so doing, he has created a better understanding of how students solve problems, as well as a better understanding of how problems should be solved and how problem solving should be taught.

For Schoenfeld, the problem solving process is ultimately a dialogue between the problem solver’s prior knowledge, his attempts, and his thoughts along the way (Schoenfeld 1982 ). As such, the solution path of a problem is an emerging and contextually dependent process. This is a departure from the predefined and contextually independent processes of Pólya’s ( 1949 ) heuristics. This can be seen in Schoenfeld’s ( 1982 ) description of a good problem solver.

To examine what accounts for expertise in problem solving, you would have to give the expert a problem for which he does not have access to a solution schema. His behavior in such circumstances is radically different from what you would see when he works on routine or familiar “non-routine” problems. On the surface his performance is no longer proficient; it may even seem clumsy. Without access to a solution schema, he has no clear indication of how to start. He may not fully understand the problem, and may simply “explore it for a while until he feels comfortable with it. He will probably try to “match” it to familiar problems, in the hope it can be transformed into a (nearly) schema-driven solution. He will bring up a variety of plausible things: related facts, related problems, tentative approaches, etc. All of these will have to be juggled and balanced. He may make an attempt solving it in a particular way, and then back off. He may try two or three things for a couple of minutes and then decide which to pursue. In the midst of pursuing one direction he may go back and say “that’s harder than it should be” and try something else. Or, after the comment, he may continue in the same direction. With luck, after some aborted attempts, he will solve the problem. (p. 32-33)

Aside from demonstrating the emergent nature of the problem solving process, this passage also brings forth two consequences of Schoenfeld’s work. The first of these is the existence of problems for which the solver does not have “access to a solution schema”. Unlike Pólya ( 1949 ), who’s heuristic is a ‘one size fits all (problems)’ heuristic, Schoenfeld acknowledges that problem solving heuristics are, in fact, personal entities that are dependent on the solver’s prior knowledge as well as their understanding of the problem at hand. Hence, the problems that a person can solve through his or her personal heuristic are finite and limited.

The second consequence that emerges from the above passage is that if a person lacks the solution schema to solve a given problem s/he may still solve the problem with the help of luck . This is an acknowledgement, if only indirectly so, of the difference between problem solving in an intentional and mechanical fashion verses problem solving in a more creative fashion, which is neither intentional nor mechanical (Pehkonen 1997 ).

1.2.7 David Perkins: Breakthrough Thinking

As mentioned, many consider a problem that can be solved by intentional and mechanical means to not be worthy of the title ‘problem’. As such, a repertoire of past experiences sufficient for dealing with such a ‘problem’ would disqualify it from the ranks of ‘problems’ and relegate it to that of ‘exercises’. For a problem to be classified as a ‘problem’, then, it must be ‘problematic’. Although such an argument is circular it is also effective in expressing the ontology of mathematical ‘problems’.

Perkins ( 2000 ) also requires problems to be problematic. His book Archimedes’ Bathtub: The Art and Logic of Breakthrough Thinking (2000) deals with situations in which the solver has gotten stuck and no amount of intentional or mechanical adherence to the principles of past experience and prior knowledge is going to get them unstuck. That is, he deals with problems that, by definition, cannot be solved through a process of design [or through the heuristics proposed by Pólya ( 1949 ) and Schoenfeld ( 1985 )]. Instead, the solver must rely on the extra-logical process of what Perkins ( 2000 ) calls breakthrough thinking .

Perkins ( 2000 ) begins by distinguishing between reasonable and unreasonable problems. Although both are solvable, only reasonable problems are solvable through reasoning. Unreasonable problems require a breakthrough in order to solve them. The problem, however, is itself inert. It is neither reasonable nor unreasonable. That quality is brought to the problem by the solver. That is, if a student cannot solve a problem by direct effort then that problem is deemed to be unreasonable for that student. Perkins ( 2000 ) also acknowledges that what is an unreasonable problem for one person is a perfectly reasonable problem for another person; reasonableness is dependent on the person.

This is not to say that, once found, the solution cannot be seen as accessible through reason. During the actual process of solving, however, direct and deductive reasoning does not work. Perkins ( 2000 ) uses several classic examples to demonstrate this, the most famous being the problem of connecting nine dots in a 3 × 3 array with four straight lines without removing pencil from paper, the solution to which is presented in Fig.  1 .

Nine dots—four lines problem and solution

To solve this problem, Perkins ( 2000 ) claims that the solver must recognize that the constraint of staying within the square created by the 3 × 3 array is a self-imposed constraint. He further claims that until this is recognized no amount of reasoning is going to solve the problem. That is, at this point in the problem solving process the problem is unreasonable. However, once this self-imposed constraint is recognized the problem, and the solution, are perfectly reasonable. Thus, the solution of an, initially, unreasonable problem is reasonable.

The problem solving heuristic that Perkins ( 2000 ) has constructed to deal with solvable, but unreasonable, problems revolves around the idea of breakthrough thinking and what he calls breakthrough problems . A breakthrough problem is a solvable problem in which the solver has gotten stuck and will require an AHA! to get unstuck and solve the problem. Perkins ( 2000 ) poses that there are only four types of solvable unreasonable problems, which he has named wilderness of possibilities , the clueless plateau , narrow canyon of exploration , and oasis of false promise . The names for the first three of these types of problems are related to the Klondike gold rush in Alaska, a time and place in which gold was found more by luck than by direct and systematic searching.

The wilderness of possibilities is a term given to a problem that has many tempting directions but few actual solutions. This is akin to a prospector searching for gold in the Klondike. There is a great wilderness in which to search, but very little gold to be found. The clueless plateau is given to problems that present the solver with few, if any, clues as to how to solve it. The narrow canyon of exploration is used to describe a problem that has become constrained in such a way that no solution now exists. The nine-dot problem presented above is such a problem. The imposed constraint that the lines must lie within the square created by the array makes a solution impossible. This is identical to the metaphor of a prospector searching for gold within a canyon where no gold exists. The final type of problem gets its name from the desert. An oasis of false promise is a problem that allows the solver to quickly get a solution that is close to the desired outcome; thereby tempting them to remain fixed on the strategy that they used to get this almost-answer. The problem is, that like the canyon, the solution does not exist at the oasis; the solution strategy that produced an almost-answer is incapable of producing a complete answer. Likewise, a desert oasis is a false promise in that it is only a reprieve from the desolation of the dessert and not a final destination.

Believing that there are only four ways to get stuck, Perkins ( 2000 ) has designed a problem solving heuristic that will “up the chances” of getting unstuck. This heuristic is based on what he refers to as “the logic of lucking out” (p. 44) and is built on the idea of introspection. By first recognizing that they are stuck, and then recognizing that the reason they are stuck can only be attributed to one of four reasons, the solver can access four strategies for getting unstuck, one each for the type of problem they are dealing with. If the reason they are stuck is because they are faced with a wilderness of possibilities they are to begin roaming far, wide, and systematically in the hope of reducing the possible solution space to one that is more manageable. If they find themselves on a clueless plateau they are to begin looking for clues, often in the wording of the problem. When stuck in a narrow canyon of possibilities they need to re-examine the problem and see if they have imposed any constraints. Finally, when in an oasis of false promise they need to re-attack the problem in such a way that they stay away from the oasis.

Of course, there are nuances and details associated with each of these types of problems and the strategies for dealing with them. However, nowhere within these details is there mention of the main difficulty inherent in introspection; that it is much easier for the solver to get stuck than it is for them to recognize that they are stuck. Once recognized, however, the details of Perkins’ ( 2000 ) heuristic offer the solver some ways for recognizing why they are stuck.

1.2.8 John Mason, Leone Burton, and Kaye Stacey: Thinking Mathematically

The work of Mason et al. in their book Thinking Mathematically ( 1982 ) also recognizes the fact that for each individual there exists problems that will not yield to their intentional and mechanical attack. The heuristic that they present for dealing with this has two main processes with a number of smaller phases, rubrics, and states. The main processes are what they refer to as specializing and generalizing. Specializing is the process of getting to know the problem and how it behaves through the examination of special instances of the problem. This process is synonymous with problem solving by design and involves the repeated oscillation between the entry and attack phases of Mason et al. ( 1982 ) heuristic. The entry phase is comprised of ‘getting started’ and ‘getting involved’ with the problem by using what is immediately known about it. Attacking the problem involves conjecturing and testing a number of hypotheses in an attempt to gain greater understanding of the problem and to move towards a solution.

At some point within this process of oscillating between entry and attack the solver will get stuck, which Mason et al. ( 1982 ) refer to as “an honourable and positive state, from which much can be learned” (p. 55). The authors dedicate an entire chapter to this state in which they acknowledge that getting stuck occurs long before an awareness of being stuck develops. They proposes that the first step to dealing with being stuck is the simple act of writing STUCK!

The act of expressing my feelings helps to distance me from my state of being stuck. It frees me from incapacitating emotions and reminds me of actions that I can take. (p. 56)

The next step is to reengage the problem by examining the details of what is known, what is wanted, what can be introduced into the problem, and what has been introduced into the problem (imposed assumptions). This process is engaged in until an AHA!, which advances the problem towards a solution, is encountered. If, at this point, the problem is not completely solved the oscillation is then resumed.

At some point in this process an attack on the problem will yield a solution and generalizing can begin. Generalizing is the process by which the specifics of a solution are examined and questions as to why it worked are investigated. This process is synonymous with the verification and elaboration stages of invention and creativity. Generalization may also include a phase of review that is similar to Pólya’s ( 1949 ) looking back.

1.2.9 Gestalt: The Psychology of Problem Solving

The Gestalt psychology of learning believes that all learning is based on insights (Koestler 1964 ). This psychology emerged as a response to behaviourism, which claimed that all learning was a response to external stimuli. Gestalt psychologists, on the other hand, believed that there was a cognitive process involved in learning as well. With regards to problem solving, the Gestalt school stands firm on the belief that problem solving, like learning, is a product of insight and as such, cannot be taught. In fact, the theory is that not only can problem solving not be taught, but also that attempting to adhere to any sort of heuristic will impede the working out of a correct solution (Krutestkii 1976 ). Thus, there exists no Gestalt problem solving heuristic. Instead, the practice is to focus on the problem and the solution rather than on the process of coming up with a solution. Problems are solved by turning them over and over in the mind until an insight, a viable avenue of attack, presents itself. At the same time, however, there is a great reliance on prior knowledge and past experiences. The Gestalt method of problem solving, then, is at the same time very different and very similar to the process of design.

Gestalt psychology has not fared well during the evolution of cognitive psychology. Although it honours the work of the unconscious mind it does so at the expense of practicality. If learning is, indeed, entirely based on insight then there is little point in continuing to study learning. “When one begins by assuming that the most important cognitive phenomena are inaccessible, there really is not much left to talk about” (Schoenfeld 1985 , p. 273). However, of interest here is the Gestalt psychologists’ claim that focus on problem solving methods creates functional fixedness (Ashcraft 1989 ). Mason et al. ( 1982 ), as well as Perkins ( 2000 ) deal with this in their work on getting unstuck.

1.2.10 Final Comments

Mathematics has often been characterized as the most precise of all sciences. Lost in such a misconception is the fact that mathematics often has its roots in the fires of creativity, being born of the extra-logical processes of illumination and intuition. Problem solving heuristics that are based solely on the processes of logical and deductive reasoning distort the true nature of problem solving. Certainly, there are problems in which logical deductive reasoning is sufficient for finding a solution. But these are not true problems. True problems need the extra-logical processes of creativity, insight, and illumination, in order to produce solutions.

Fortunately, as elusive as such processes are, there does exist problem solving heuristics that incorporate them into their strategies. Heuristics such as those by Perkins ( 2000 ) and Mason et al. ( 1982 ) have found a way of combining the intentional and mechanical processes of problem solving by design with the extra-logical processes of creativity, illumination, and the AHA!. Furthermore, they have managed to do so without having to fully comprehend the inner workings of this mysterious process.

1.3 Digital Technologies and Mathematical Problem Solving—Luz Manuel Santos-Trigo

Mathematical problem solving is a field of research that focuses on analysing the extent to which problem solving activities play a crucial role in learners’ understanding and use of mathematical knowledge. Mathematical problems are central in mathematical practice to develop the discipline and to foster students learning (Pólya 1945 ; Halmos 1994 ). Mason and Johnston-Wilder ( 2006 ) pointed out that “The purpose of a task is to initiate mathematically fruitful activity that leads to a transformation in what learners are sensitized to notice and competent to carry out” (p. 25). Tasks are essential for learners to elicit their ideas and to engage them in mathematical thinking. In a problem solving approach, what matters is the learners’ goals and ways to interact with the tasks. That is, even routine tasks can be a departure point for learners to extend initial conditions and transform them into some challenging activities.

Thus, analysing and characterizing ways in which mathematical problems are formulated (Singer et al. 2015 ) and the process involved in pursuing and solving those problems generate important information to frame and structure learning environments to guide and foster learners’ construction of mathematical concepts and problem solving competences (Santos-Trigo 2014 ). Furthermore, mathematicians or discipline practitioners have often been interested in unveiling and sharing their own experience while developing the discipline. As a results, they have provided valuable information to characterize mathematical practices and their relations to what learning processes of the discipline entails. It is recognized that the work of Pólya ( 1945 ) offered not only bases to launch several research programs in problem solving (Schoenfeld 1992 ; Mason et al. 1982 ); but also it became an essential resource for teachers to orient and structure their mathematical lessons (Krulik and Reys 1980 ).

1.3.1 Research Agenda

A salient feature of a problem solving approach to learn mathematics is that teachers and students develop and apply an enquiry or inquisitive method to delve into mathematical concepts and tasks. How are mathematical problems or concepts formulated? What types of problems are important for teachers/learners to discuss and engage in mathematical reasoning? What mathematical processes and ways of reasoning are involved in understanding mathematical concepts and solving problems? What are the features that distinguish an instructional environment that fosters problem-solving activities? How can learners’ problem solving competencies be assessed? How can learners’ problem solving competencies be characterized and explained? How can learners use digital technologies to understand mathematics and to develop problem-solving competencies? What ways of reasoning do learners construct when they use digital technologies in problem solving approaches? These types of questions have been important in the problem solving research agenda and delving into them has led researchers to generate information and results to support and frame curriculum proposals and learning scenarios. The purpose of this section is to present and discuss important themes that emerged in problem solving approaches that rely on the systematic use of several digital technologies.

In the last 40 years, the accumulated knowledge in the problem solving field has shed lights on both a characterization of what mathematical thinking involves and how learners can construct a robust knowledge in problem solving environments (Schoenfeld 1992 ). In this process, the field has contributed to identify what types of transformations traditional learning scenarios might consider when teachers and students incorporate the use of digital technologies in mathematical classrooms. In this context, it is important to briefly review what main themes and developments the field has addressed and achieved during the last 40 years.

1.3.2 Problem Solving Developments

There are traces of mathematical problems and solutions throughout the history of civilization that explain the humankind interest for identifying and exploring mathematical relations (Kline 1972 ). Pólya ( 1945 ) reflects on his own practice as a mathematician to characterize the process of solving mathematical problems through four main phases: Understanding the problem, devising a plan, carrying out the plan, and looking back. Likewise, Pólya ( 1945 ) presents and discusses the role played by heuristic methods throughout all problem solving phases. Schoenfeld ( 1985 ) presents a problem solving research program based on Pólya’s ( 1945 ) ideas to investigate the extent to which problem solving heuristics help university students to solve mathematical problems and to develop a way of thinking that shows consistently features of mathematical practices. As a result, he explains the learners’ success or failure in problem solving activities can be characterized in terms their mathematical resources and ways to access them, cognitive and metacognitive strategies used to represent and explore mathematical tasks, and systems of beliefs about mathematics and solving problems. In addition, Schoenfeld ( 1992 ) documented that heuristics methods as illustrated in Pólya’s ( 1945 ) book are ample and general and do not include clear information and directions about how learners could assimilate, learn, and use them in their problem solving experiences. He suggested that students need to discuss what it means, for example, to think of and examining special cases (one important heuristic) in finding a closed formula for series or sequences, analysing relationships of roots of polynomials, or focusing on regular polygons or equilateral/right triangles to find general relations about these figures. That is, learners need to work on examples that lead them to recognize that the use of a particular heuristic often involves thinking of different type of cases depending on the domain or content involved. Lester and Kehle ( 2003 ) summarize themes and methodological shifts in problem solving research up to 1995. Themes include what makes a problem difficult for students and what it means to be successful problem solvers; studying and contrasting experts and novices’ problem solving approaches; learners’ metacognitive, beliefs systems and the influence of affective behaviours; and the role of context; and social interactions in problem solving environments. Research methods in problem solving studies have gone from emphasizing quantitative or statistical design to the use of cases studies and ethnographic methods (Krutestkii ( 1976 ). Teaching strategies also evolved from being centred on teachers to the active students’ engagement and collaboration approaches (NCTM 2000 ). Lesh and Zawojewski ( 2007 ) propose to extend problem solving approaches beyond class setting and they introduce the construct “model eliciting activities” to delve into the learners’ ideas and thinking as a way to engage them in the development of problem solving experiences. To this end, learners develop and constantly refine problem-solving competencies as a part of a learning community that promotes and values modelling construction activities. Recently, English and Gainsburg ( 2016 ) have discussed the importance of modeling eliciting activities to prepare and develop students’ problem solving experiences for 21st Century challenges and demands.

Törner et al. ( 2007 ) invited mathematics educators worldwide to elaborate on the influence and developments of problem solving in their countries. Their contributions show a close relationship between countries mathematical education traditions and ways to frame and implement problem solving approaches. In Chinese classrooms, for example, three instructional strategies are used to structure problem solving lessons: one problem multiple solutions , multiple problems one solution , and one problem multiple changes . In the Netherlands, the realistic mathematical approach permeates the students’ development of problem solving competencies; while in France, problem solving activities are structured in terms of two influential frameworks: The theory of didactical situations and anthropological theory of didactics.

In general, problem solving frameworks and instructional approaches came from analysing students’ problem solving experiences that involve or rely mainly on the use of paper and pencil work. Thus, there is a need to re-examined principles and frameworks to explain what learners develop in learning environments that incorporate systematically the coordinated use of digital technologies (Hoyles and Lagrange 2010 ). In this perspective, it becomes important to briefly describe and identify what both multiple purpose and ad hoc technologies can offer to the students in terms of extending learning environments and representing and exploring mathematical tasks. Specifically, a task is used to identify features of mathematical reasoning that emerge through the use digital technologies that include both mathematical action and multiple purpose types of technologies.

1.3.3 Background

Digital technologies are omnipresent and their use permeates and shapes several social and academic events. Mobile devices such as tablets or smart phones are transforming the way people communicate, interact and carry out daily activities. Churchill et al. ( 2016 ) pointed out that mobile technologies provide a set of tools and affordances to structure and support learning environments in which learners continuously interact to construct knowledge and solve problems. The tools include resources or online materials, efficient connectivity to collaborate and discuss problems, ways to represent, explore and store information, and analytical and administration tools to management learning activities. Schmidt and Cohen ( 2013 ) stated that nowadays it is difficult to imagine a life without mobile devices, and communication technologies are playing a crucial role in generating both cultural and technical breakthroughs. In education, the use of mobile artefacts and computers offers learners the possibility of continuing and extending peers and groups’ mathematical discussions beyond formal settings. In this process, learners can also consult online materials and interact with experts, peers or more experienced students while working on mathematical tasks. In addition, dynamic geometry systems (GeoGebra) provide learners a set of affordances to represent and explore dynamically mathematical problems. Leung and Bolite-Frant ( 2015 ) pointed out that tools help activate an interactive environment in which teachers and students’ mathematical experiences get enriched. Thus, the digital age brings new challenges to the mathematics education community related to the changes that technologies produce to curriculum, learning scenarios, and ways to represent, explore mathematical situations. In particular, it is important to characterize the type of reasoning that learners can develop as a result of using digital technologies in their process of learning concepts and solving mathematical problems.

1.3.4 A Focus on Mathematical Tasks

Mathematical tasks are essential elements for engaging learners in mathematical reasoning which involves representing objects, identifying and exploring their properties in order to detect invariants or relationships and ways to support them. Watson and Ohtani ( 2015 ) stated that task design involves discussions about mathematical content and students’ learning (cognitive perspective), about the students’ experiences to understand the nature of mathematical activities; and about the role that tasks played in teaching practices. In this context, tasks are the vehicle to present and discuss theoretical frameworks for supporting the use of digital technology, to analyse the importance of using digital technologies in extending learners’ mathematical discussions beyond formal settings, and to design ways to foster and assess the use of technologies in learners’ problem solving environments. In addition, it is important to discuss contents, concepts, representations and strategies involved in the process of using digital technologies in approaching the tasks. Similarly, it becomes essential to discuss what types of activities students will do to learn and solve the problems in an environment where the use of technologies fosters and values the participation and collaboration of all students. What digital technologies are important to incorporate in problem solving approaches? Dynamic Geometry Systems can be considered as a milestone in the development of digital technologies. Objects or mathematical situations can be represented dynamically through the use of a Dynamic Geometry System and learners or problem solvers can identify and examine mathematical relations that emerge from moving objects within the dynamic model (Moreno-Armella and Santos-Trigo 2016 ).

Leung and Bolite-Frant ( 2015 ) stated that “dynamic geometry software can be used in task design to cover a large epistemic spectrum from drawing precise robust geometrical figures to exploration of new geometric theorems and development of argumentation discourse” (p. 195). As a result, learners not only need to develop skills and strategies to construct dynamic configuration of problems; but also ways of relying on the tool’s affordances (quantifying parameters or objects attributes, generating loci, graphing objects behaviours, using sliders, or dragging particular elements within the configuration) in order to identify and support mathematical relations. What does it mean to represent and explore an object or mathematical situation dynamically?

A simple task that involves a rhombus and its inscribed circle is used to illustrate how a dynamic representation of these objects and embedded elements can lead learners to identify and examine mathematical properties of those objects in the construction of the configuration. To this end, learners are encouraged to pose and pursue questions to explain the behaviours of parameters or attributes of the family of objects that is generated as a result of moving a particular element within the configuration.

1.3.5 A Task: A Dynamic Rhombus

Figure  2 represents a rhombus APDB and its inscribed circle (O is intersection of diagonals AD and BP and the radius of the inscribed circle is the perpendicular segment from any side of the rhombus to point O), vertex P lies on a circle c centred at point A. Circle c is only a heuristic to generate a family of rhombuses. Thus, point P can be moved along circle c to generate a family of rhombuses. Indeed, based on the symmetry of the circle it is sufficient to move P on the semicircle B’CA to draw such a family of rhombuses.

A dynamic construction of a rhombus

1.3.6 Posing Questions

A goal in constructing a dynamic model or configuration of problems is always to identify and explore mathematical properties and relations that might result from moving objects within the model. How do the areas of both the rhombus and the inscribed circle behave when point P is moved along the arc B’CB? At what position of point P does the area of the rhombus or inscribed circle reach the maximum value? The coordinates of points S and Q (Fig.  3 ) are the x -value of point P and as y -value the corresponding area values of rhombus ABDP and the inscribed circle respectively. Figure  2 shows the loci of points S and Q when point P is moved along arc B’CB. Here, finding the locus via the use of GeoGebra is another heuristic to graph the area behaviour without making explicit the algebraic model of the area.

Graphic representation of the area variation of the family of rhombuses and inscribed circles generated when P is moved through arc B’CB

The area graphs provide information to visualize that in that family of generated rhombuses the maximum area value of the inscribed circle and rhombus is reached when the rhombus becomes a square (Fig.  4 ). That is, the controlled movement of particular objects is an important strategy to analyse the area variation of the family of rhombuses and their inscribed circles.

Visualizing the rhombus and the inscribed circle with maximum area

It is important to observe the identification of points P and Q in terms of the position of point P and the corresponding areas and the movement of point P was sufficient to generate both area loci. That is, the graph representation of the areas is achieved without having an explicit algebraic expression of the area variation. Clearly, the graphic representations provide information regarding the increasing or decreasing interval of both areas; it is also important to explore what properties both graphic representations hold. The goal is to argue that the area variation of the rhombus represents an ellipse and the area of the inscribed circle represents a parabola. An initial argument might involve selecting five points on each locus and using the tool to draw the corresponding conic section (Fig.  5 ). In this case, the tool affordances play an important role in generating the graphic representation of the areas’ behaviours and in identifying properties of those representations. In this context, the use of the tool can offer learners the opportunity to problematize (Santos-Trigo 2007 ) a simple mathematical object (rhombus) as a means to search for mathematical relations and ways to support them.

Drawing the conic section that passes through five points

1.3.7 Looking for Different Solutions Methods

Another line of exploration might involve asking for ways to construct a rhombus and its inscribed circle: Suppose that the side of the rhombus and the circle are given, how can you construct the rhombus that has that circle inscribed? Figure  6 shows the given data, segment A 1 B 1 and circle centred at O and radius OD. The initial goal is to draw the circle tangent to the given segment. To this end, segment AB is congruent to segment A 1 B 1 and on this segment a point P is chosen and a perpendicular to segment AB that passes through point P is drawn. Point C is on this perpendicular and the centre of a circle with radius OD and h is the perpendicular to line PC that passes through point C. Angle ACB changes when point P is moved along segment AB and point E and F are the intersection of line h and the circle with centre M the midpoint of AB and radius MA (Fig.  6 ).

Drawing segment AB tangent to the given circle

Figure  7 a shows the right triangle AFB as the base to construct the rhombus and the inscribed circle and Fig.  7 b shows the second solution based on triangle AEB.

a Drawing the rhombus and the inscribed circle. b Drawing the second solution

Another approach might involve drawing the given circle centred at the origin and the segment as EF with point E on the y-axis. Line OC is perpendicular to segment EF and the locus of point C when point E moves along the y-axis intersects the given circle (Fig.  8 a, b). Both figures show two solutions to draw the rhombus that circumscribe the given circle.

a and b Another solution that involves finding a locus of point C

In this example, the GeoGebra affordances not only are important to construct a dynamic model of the task; but also offer learners and opportunity to explore relations that emerge from moving objects within the model. As a result, learners can rely on different concepts and strategies to solve the tasks. The idea in presenting this rhombus task is to illustrate that the use of a Dynamic Geometry System provides affordances for learners to construct dynamic representation of mathematical objects or problems, to move elements within the representation to pose questions or conjectures to explain invariants or patterns among involved parameters; to search for arguments to support emerging conjectures, and to develop a proper language to communicate results.

1.3.8 Looking Back

Conceptual frameworks used to explain learners’ construction of mathematical knowledge need to capture or take into account the different ways of reasoning that students might develop as a result of using a set of tools during the learning experiences. Figure  9 show some digital technologies that learners can use for specific purpose at the different stages of problem solving activities.

The coordinated use of digital tools to engage learners in problem solving experiences

The use of a dynamic system (GeoGebra) provides a set of affordances for learners to conceptualize and represent mathematical objects and tasks dynamically. In this process, affordances such as moving objects orderly (dragging), finding loci of objects, quantifying objects attributes (lengths, areas, angles, etc.), using sliders to vary parameters, and examining family of objects became important to look for invariance or objects relationships. Likewise, analysing the parameters or objects behaviours within the configuration might lead learners to identify properties to support emerging mathematical relations. Thus, with the use of the tool, learners might conceptualize mathematical tasks as an opportunity for them to engage in mathematical activities that include constructing dynamic models of tasks, formulating conjectures, and always looking for different arguments to support them. Similarly, learners can use an online platform to share their ideas, problem solutions or questions in a digital wall and others students can also share ideas or solution methods and engaged in mathematical discussions that extend mathematical classroom activities.

1.4 Problem Posing: An Overview for Further Progress—Uldarico Malaspina Jurado

Problem posing and problem solving are two essential aspects of the mathematical activity; however, researchers in mathematics education have not emphasized their attention on problem posing as much as problem solving. In that sense, due to its importance in the development of mathematical thinking in students since the first grades, we agree with Ellerton’s statement ( 2013 ): “for too long, successful problem solving has been lauded as the goal; the time has come for problem posing to be given a prominent but natural place in mathematics curricula and classrooms” (pp. 100–101); and due to its importance in teacher training, with Abu-Elwan’s statement ( 1999 ):

While teacher educators generally recognize that prospective teachers require guidance in mastering the ability to confront and solve problems, what is often overlooked is the critical fact that, as teachers, they must be able to go beyond the role as problem solvers. That is, in order to promote a classroom situation where creative problem solving is the central focus, the practitioner must become skillful in discovering and correctly posing problems that need solutions. (p. 1)

Scientists like Einstein and Infeld ( 1938 ), recognized not only for their notable contributions in the fields they worked, but also for their reflections on the scientific activity, pointed out the importance of problem posing; thus it is worthwhile to highlight their statement once again:

The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skills. To raise new questions, new possibilities, to regard old questions from a new angle, requires creative imagination and marks real advance in science. (p. 92)

Certainly, it is also relevant to remember mathematician Halmos’s statement ( 1980 ): “I do believe that problems are the heart of mathematics, and I hope that as teachers (…) we will train our students to be better problem posers and problem solvers than we are” (p. 524).

An important number of researchers in mathematics education has focused on the importance of problem posing, and we currently have numerous, very important publications that deal with different aspects of problem posing related to the mathematics education of students in all educational levels and to teacher training.

1.4.1 A Retrospective Look

Kilpatrick ( 1987 ) marked a historical milestone in research related to problem posing and points out that “problem formulating should be viewed not only as a goal of instruction but also as a means of instruction” (Kilpatrick 1987 , p. 123); and he also emphasizes that, as part of students’ education, all of them should be given opportunities to live the experience of discovering and posing their own problems. Drawing attention to the few systematic studies on problem posing performed until then, Kilpatrick contributes defining some aspects that required studying and investigating as steps prior to a theoretical building, though he warns, “attempts to teach problem-formulating skills, of course, need not await a theory” (p. 124).

Kilpatrick refers to the “Source of problems” and points out how virtually all problems students solve have been posed by another person; however, in real life “many problems, if not most, must be created or discovered by the solver, who gives the problem an initial formulation” (p. 124). He also points out that problems are reformulated as they are being solved, and he relates this to investigation, reminding us what Davis ( 1985 ) states that, “what typically happens in a prolonged investigation is that problem formulation and problem solution go hand in hand, each eliciting the other as the investigation progresses” (p. 23). He also relates it to the experiences of software designers, who formulate an appropriate sequence of sub-problems to solve a problem. He poses that a subject to be examined by teachers and researchers “is whether, by drawing students’ attention to the reformulating process and given them practice in it, we can improve their problem solving performance” (p. 130). He also points out that problems may be a mathematical formulation as a result of exploring a situation and, in that sense, “school exercises in constructing mathematical models of a situation presented by the teacher are intended to provide students with experiences in formulating problems.” (p. 131).

Another important section of Kilpatrick’s work ( 1987 ) is Processes of Problem Formulating , in which he considers association, analogy, generalization and contradiction. He believes the use of concept maps to represent concept organization, as cognitive scientists Novak and Gowin suggest, might help to comprehend such concepts, stimulate creative thinking about them, and complement the ideas Brown and Walter ( 1983 ) give for problem posing by association. Further, in the section “Understanding and developing problem formulating abilities”, he poses several questions, which have not been completely answered yet, like “Perhaps the central issue from the point of view of cognitive science is what happens when someone formulates the problem? (…) What is the relation between problem formulating, problem solving and structured knowledge base? How rich a knowledge base is needed for problem formulating? (…) How does experience in problem formulating add to knowledge base? (…) What metacognitive processes are needed for problem formulating?”

It is interesting to realize that some of these questions are among the unanswered questions proposed and analyzed by Cai et al. ( 2015 ) in Chap. 1 of the book Mathematical Problem Posing (Singer et al. 2015 ). It is worth stressing the emphasis on the need to know the cognitive processes in problem posing, an aspect that Kilpatrick had already posed in 1987, as we just saw.

1.4.2 Researches and Didactic Experiences

Currently, there are a great number of publications related to problem posing, many of which are research and didactic experiences that gather the questions posed by Kilpatrick, which we just commented. Others came up naturally as reflections raised in the framework of problem solving, facing the natural requirement of having appropriate problems to use results and suggestions of researches on problem solving, or as a response to a thoughtful attitude not to resign to solving and asking students to solve problems that are always created by others. Why not learn and teach mathematics posing one’s own problems?

1.4.3 New Directions of Research

Singer et al. ( 2013 ) provides a broad view about problem posing that links problem posing experiences to general mathematics education; to the development of abilities, attitudes and creativity; and also to its interrelation with problem solving, and studies on when and how problem-solving sessions should take place. Likewise, it provides information about research done regarding ways to pose new problems and about the need for teachers to develop abilities to handle complex situations in problem posing contexts.

Singer et al. ( 2013 ) identify new directions in problem posing research that go from problem-posing task design to the development of problem-posing frameworks to structure and guide teachers and students’ problem posing experiences. In a chapter of this book, Leikin refers to three different types of problem posing activities, associated with school mathematics research: (a) problem posing through proving; (b) problem posing for investigation; and (c) problem posing through investigation. This classification becomes evident in the problems posed in a course for prospective secondary school mathematics teachers by using a dynamic geometry environment. Prospective teachers posed over 25 new problems, several of which are discussed in the article. The author considers that, by developing this type of problem posing activities, prospective mathematics teachers may pose different problems related to a geometric object, prepare more interesting lessons for their students, and thus gradually develop their mathematical competence and their creativity.

1.4.4 Final Comments

This overview, though incomplete, allows us to see a part of what problem posing experiences involve and the importance of this area in students mathematical learning. An important task is to continue reflecting on the questions posed by Kilpatrick ( 1987 ), as well as on the ones that come up in the different researches aforementioned. To continue progressing in research on problem posing and contribute to a greater consolidation of this research line, it will be really important that all mathematics educators pay more attention to problem posing, seek to integrate approaches and results, and promote joint and interdisciplinary works. As Singer et al. ( 2013 ) say, going back to Kilpatrick’s proposal ( 1987 ),

Problem posing is an old issue. What is new is the awareness that problem posing needs to pervade the education systems around the world, both as a means of instruction (…) and as an object of instruction (…) with important targets in real-life situations. (p. 5)

Although it can be argued that there is a difference between creativity, discovery, and invention (see Liljedahl and Allan 2014 ) for the purposes of this book these will be assumed to be interchangeable.

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Further Reading

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Liljedahl, P., Santos-Trigo, M., Malaspina, U., Bruder, R. (2016). Problem Solving in Mathematics Education. In: Problem Solving in Mathematics Education. ICME-13 Topical Surveys. Springer, Cham. https://doi.org/10.1007/978-3-319-40730-2_1

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Features of good problem solving tasks for learning mathematics

To develop higher-order thinking skills (HOTS) the mind needs to engage in higher-order learning task (HOLT). A good task for developing higher-order thinking skills is a problem solving task. But not all problems are created equal. Some problems are best suited for evaluating learning while others are best suited for assessing learning that would inform teaching. This post is about the second set of problems.The difference between these two sets of problems is not the content and skills needed to solve them but the way they are constructed.

What are the features of a good problem solving task for learning mathematics?

• It uses contexts familiar to the students
• What is problematic is the mathematics rather than the aspect of the situation
• It encourages students to use intuitive solutions as well as knowledge and skills they already possess
• The task can have several solutions
• It challenges students to use the strategy that would highlight the depth of their understanding of the concept involved
• It allows students to show the connections they have made between the concepts they have learned

It is this kind of problem solving task that is used in the strategy Teaching through Problem Solving (T t PS) which I described in the previous post. Here is a sample task:

Students solutions to the task can be used to teach area of polygons, kinds of polygons, preserving area, and meaning of algebraic expression. You can use the task to facilitate students construction of knowledge about adding, subtracting, multiplying and dividing algebraic expressions. Yes, you read it right. This is a good problem solving task for introducing operations with algebraic expression through problem solving! The problem above is also an example of a mathematical tasks that links algebra and geometry. Good mathematics teaching always links concepts.

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• Negative numbers and coordinate plane
• Ratios, rates, proportions
• Equations, expressions, and inequalities
• Exponents, radicals, and scientific notation
• Foundations
• Algebraic expressions
• Linear equations and inequalities
• Graphing lines and slope
• Expressions with exponents
• Quadratics and polynomials
• Equations and geometry
• Algebra foundations
• Solving equations & inequalities
• Working with units
• Linear equations & graphs
• Forms of linear equations
• Inequalities (systems & graphs)
• Absolute value & piecewise functions
• Exponents & radicals
• Exponential growth & decay
• Quadratics: Multiplying & factoring
• Quadratic functions & equations
• Irrational numbers
• Performing transformations
• Transformation properties and proofs
• Right triangles & trigonometry
• Non-right triangles & trigonometry (Advanced)
• Analytic geometry
• Conic sections
• Solid geometry
• Polynomial arithmetic
• Complex numbers
• Polynomial factorization
• Polynomial division
• Polynomial graphs
• Rational exponents and radicals
• Exponential models
• Transformations of functions
• Rational functions
• Trigonometric functions
• Non-right triangles & trigonometry
• Trigonometric equations and identities
• Analyzing categorical data
• Displaying and comparing quantitative data
• Summarizing quantitative data
• Modeling data distributions
• Exploring bivariate numerical data
• Study design
• Probability
• Counting, permutations, and combinations
• Random variables
• Sampling distributions
• Confidence intervals
• Significance tests (hypothesis testing)
• Two-sample inference for the difference between groups
• Inference for categorical data (chi-square tests)
• Advanced regression (inference and transforming)
• Analysis of variance (ANOVA)
• Scatterplots
• Data distributions
• Two-way tables
• Binomial probability
• Normal distributions
• Displaying and describing quantitative data
• Inference comparing two groups or populations
• Chi-square tests for categorical data
• More on regression
• Prepare for the 2020 AP®︎ Statistics Exam
• AP®︎ Statistics Standards mappings
• Polynomials
• Composite functions
• Probability and combinatorics
• Limits and continuity
• Derivatives: definition and basic rules
• Derivatives: chain rule and other advanced topics
• Applications of derivatives
• Analyzing functions
• Parametric equations, polar coordinates, and vector-valued functions
• Applications of integrals
• Differentiation: definition and basic derivative rules
• Differentiation: composite, implicit, and inverse functions
• Contextual applications of differentiation
• Applying derivatives to analyze functions
• Integration and accumulation of change
• Applications of integration
• AP Calculus AB solved free response questions from past exams
• AP®︎ Calculus AB Standards mappings
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• AP Calculus BC solved exams
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Frequently Asked Questions about Khan Academy and Math Worksheets

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• 15 hours ago

10 Math Tasks for the Beginning of the Year

I have a rule about the first day of school: always do some math. No, that doesn't mean you have to start Lesson 1.1 as soon as students walk through the door, but it does mean that you should give your students a preview of the kind of thinking, reasoning, puzzling, and sense-making that they'll be doing in your class this year. Ideally, students will be so highly engaged that they barely even recognize they're doing math -- and certainly not the kind of math they're used to in school.

I also have an inordinate appreciation for what I call "interesting problems". These are tasks that use mathematical thinking and strategy, but don't require specific content knowledge like the formula for the equation of a circle or knowing what a composite function does. They are highly accessible, highly engaging, and have multiple solution strategies. The task itself can be explained in a few sentences and students can work on them for 20 minutes or 2 hours, depending on how far they want to take it. I scour the internet for tasks like these and have been collecting them for YEARS on my computer. I decided this was the year to bring them to the light and share them with the Math Medic community.

These tasks don't require formal content knowledge, but they do help students engage in the mathematical practices and develop mathematical habits of mind, such as:

Looking for and making use of structure

Representing one's thinking

Working systematically

Visualizing

Developing a convincing argument

Conjecturing and generalizing

While I've curated this list with high school students in mind, many of these tasks could be done with middle schoolers or even with adults. The inspiration for these questions came from all over this great big internet, but have been adapted and reformatted for classroom use. So, without further ado, here are my (current) top 10 "interesting problems" to do on the first day of school.

10 Interesting Problems

The Shopping Cart Task

A linear context in a LOT of disguise. Many solution strategies and great opportunities for representing one's thinking with a model or visual.

The Locker Problem

This one is set up with multiple parts providing lots of natural extensions. Thinking about a number's properties is key to this task! Make sure to print the 100s chart that is on page two on a separate sheet of paper. You can offer it to everyone or as an optional support.

A Leg to Stand On

Loads of solution strategies on this one as well. Your teacher brain might scream system of equations with 4 variables, but you'll be surprised at the intuitive solutions your students find to solve this problem.

To Run or Not to Run?

Perfect after a summer of olympics. Students deal with rates in this problem, which is an important concept for any age group and relevant for any math course.

Four 4's

This one's been famous for a long time but I'm sharing it anyway because students do great with it!

Page Turner

This one and the next two all encourage students to think systematically. There's a brute force solution but making use of structure will illuminate an easier way.

How Many 7's?

This is a good intro to thinking systematically and has a nice extension. I would use this in an Algebra 1 or Geometry course.

Oddball Numbers

This one is very difficult, so we recommend saving it for your Precalc or above courses.

This one is the most recent in my collection and I'm still thinking about the extension part!

Toothpick Challenge

I've often done this one with Geometry students because of the shapes and visual reasoning components.

Editable versions of these tasks can be found in this Google Drive folder .

How to use these in your classroom:

Pick ONE task for students to work on. We don't recommend giving multiple tasks back-to-back because it can start to feel like a worksheet, rather than a puzzle.

Solve the problem yourself first! We are purposefully not giving solutions here , so make sure you've wrestled with the problem yourself before handing it out to students.

Have students work on these in groups of 2, 3, or 4. Make sure they have enough materials available to hash out their ideas and represent their strategies. These are great to do on vertical non-permanent surfaces or poster paper.

Decide how long you will let students work. If doing this on the first day of school, we recommend about 20 minutes. If students don't have a solution by then, that is totally fine. A surprising number of them will keep thinking about the problem throughout the day or even at home.

Be ready with some extensions for groups who finish early, but make sure they understand what "done" means. Have they clearly communicated their strategy? Have they convinced themselves and others that their strategy will hold up? If giving an extension, make sure it's related to the given task, not just a different task. It's important that students are challenged in the depth of their reasoning, not in the quantity of problems.

If you're looking for more tasks like these, I highly recommend the NRICH site from the University of Cambridge.

• Math Medic Core

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Teaching methods: Engaging students with problem solving tasks in maths

Thanks for downloading this podcast from Teacher magazine. I’m Dominique Russell.

Research shows that challenging problem solving tasks have a positive impact on student learning, but there is little evidence on student attitudes towards problem solving when it comes to doing this in the maths classroom.

My guest today is education consultant at Love Maths , Michael Minas. He works in primary schools to help improve learning in mathematics through professional development, classroom modelling and work with parents.

His areas of interest include problem solving and student engagement, and during his time working in primary school maths classrooms, he’s noticed anecdotally that students respond really positively to being presented with challenging problem solving tasks.

So, to formally investigate this, he conducted a study to assess student attitudes towards problem solving in maths alongside Dr James Russo from Monash University. This study focused on 52 students in two classrooms – a Year 3 and 4 class and a Year 5 and 6 class – in a primary school in Melbourne. Michael led a number of lessons in each classroom which presented challenging problem solving tasks to students. The classroom teachers observed these lessons, and then led these same tasks with the students. The lesson structure used was launch-explore-discuss/summarise, which Michael will go over in more detail in the episode.

After these lessons, the students completed a questionnaire to assess their opinions on the task. The results found that three-quarters of students reported unambiguously positive attitudes towards problem solving, the others were ambivalent, and no student expressed a negative attitude.

So, if you’re interested in implementing challenging problem solving tasks in your classroom, keep listening to hear Michael explain in detail the structure of these tasks, and what elements students enjoyed most. Let’s jump in.

Dominique Russell: Thanks for joining me Michael. I just thought it would be good to get a bit of background on the work you’re doing at the moment to start things off and why this research was important for you to conduct?

Michael Minas: Yeah I guess a lot of my work at the moment is in classrooms and one of the things that a lot of the schools are interested in is trying to get more problem solving happening in their mathematics classrooms. So a lot of the work that I do is in classrooms modelling problem solving lessons, working with teachers to sort of develop their sort of approach, their level of comfort with that style of teaching.

And so for me this was really interesting because, you know, I know anecdotally through my sort of experience with working with hundreds and hundreds of students, that I can see the positive responses. But, you know, it’s obviously an area that hasn’t had a lot of research done into it, so it’s good to be able to have, you know, the start of looking at it in a more formal way, of how do students actually feel in these types of lessons? What’s the experience like for them?

DR: And so the research obviously looks at two classrooms in particular in a primary school in Melbourne, looking at those middle years in primary school, which like you say, hasn’t really been looked at in much detail in the literature. So can you describe for me a bit about the school context of this particular school that you were doing the research in?

MM: Yeah, so I mean it’s a typical sort of primary school. It wasn’t, it didn’t have sort of anything outstanding in terms of the cohort of students, the size of the school – you know, 300 kids – it was a pretty sort of, demographically, a regular mix of students.

In terms of mathematics, it was philosophically quite a traditional environment for students to work in with a lot of sort of teacher directed work – you know, ‘I’ll show you how to do it, and then you go back to your tables and you reproduce what I’ve put on the board and maybe answer a series of questions using the approach that I’ve shown you’. So this style of lesson and learning was quite different for both the staff and the students at the school.

DR: And so obviously this style of learning that you exposed them to was received very positively from the students involved which we’ll talk about in a bit more detail soon, but I’m interested then in what the students opinions were of maths before this problem solving task was introduced to them. Do you have any concept of how they viewed maths in general? Did they enjoy it or were they enthusiastic about it?

MM: Yeah, so we had a couple things. So, you know, fairly informal, but when I arrived there one of the first things I did was I did some surveys with all of the students from Year 3-6. And the surveys were around their attitude to maths and also their self-perception (so, how they saw themselves as maths learners) and there were some really clear negative trends there.

So that was a starting point, you know, working with the leadership in the school to say, there are some issues here and you know, you can clearly sort of see that there are some issues here from the survey data.

But beyond that, I mean, anecdotally, my very first day at the school, I distinctly remember this. I was walking into one of the rooms at the school and a little Grade 3 girl said to me ‘oh, who are you?’ and I said, you know, ‘I’m Michael, I’m going to be here, I’m going to be working with you guys on maths’ and whatever. And her friend that was sitting with her, who wasn’t part of the conversation, inserted herself into the conversation to say, ‘oh, we hate maths. Both of us hate maths.’ Like really wanted to make a point of letting me know that she hated maths.

So that type of interaction was probably the most memorable, but I had lots of those types of interactions where people said ‘oh, you’re working with maths? Yeah I don’t like learning maths. I’m not interested in maths. I hate maths. Maths is my least favourite subject.’

So I had lots of those interactions with the kids and, you know, with that girl on the very first day and I said to her ‘well, you know, hopefully if we have this same conversation in November, that you will have shifted the way that you see maths. But that’s my job to do that, that’s not your job.’

DR: And so can you talk me through really the structure of these problem solving tasks that you led in the classrooms? Because I know you were leading them for a little while and the classroom teacher was observing the lessons that you were conducting. So what’s the structure of these tasks?

MM: Yeah, so one of the things is that the structure is a very, sort of central feature of this approach. And the idea is that, that structure is meant to be very predictable for both the staff and for the students.

You know, so we’d start with a warm up activity and the central idea of that is that you want that to be an engaging warm up to have the kids starting the lesson with, you know, a lot of energy and enthusiasm. And that would be followed by the launch of the problem. And, ideally, most problems, we want them to be launched with some of narrative link, some sort of connection to the real world. And we want that to be done in a concise way.

So the idea there is it’s not like a mini lesson of ‘let me get up the front and tell you everything I know about division for the next 15 minutes’. The idea is that we’re giving them a task that – maybe the task lends itself to multiplicative thinking and division, but we’re leaving space for the students to approach the task from their own perspective. So that, I will say to teachers ideally I want that launch to be sort of somewhere around the five minute mark. And for a lot of classroom teachers that’s a challenge, that sort of directly conflicts with the way they’re currently taking their maths lessons.

And then by extension, the shorter that launch time is, the more time the students have to be exploring, engaged with the task. So in order for, you know, students to stay working on a task for 35, 40 minutes, the task needs to be challenging, it needs to be cognitively engaging for them.

And so that explore time starts with five minutes of silent, independent work. And it’s really important that it is silent and is independent. And then from there, I’m a big advocate for actively encouraging collaboration in the classroom. So, not just saying ‘if you want to work with someone, you can’, but actively encouraging the kids; say: ‘hey, why don’t you go over and talk to Megan and see what she’s doing because she’s got some similar thoughts to you, but she’s approaching it a bit of a different way’.

And then the lessons will always finish with some sort of summary of what we’ve done and that again is student-centred. So the idea is that we’re (myself or whoever the teacher is) is looking for student examples to sort of showcase at the end of the lesson to say: ‘hey, you know, talk to me Dominique about what you’ve done’, and getting you to explain your thinking, but being really strategic about it who you select. So it’s not like ‘everyone come to the floor, okay, who’d like to share their work?’ and the same three kids put their hands up every day. It’s you as a teacher being really strategic about who you select and why.

DR: And so something that I found really interesting in the report, just as a bit of an example of how this structure plays out, the example of the chessboard tournament problem. So, the problem was launched with a short story about a family holiday and there was a big chessboard where they were staying. So you displayed a photo of one of the children playing on the chess board at the front of the classroom, and then you gave the task, which was if six children wanted to have a round robin tournament, how many games would need to be played? And then you had a prompt, which asked students to draw a diagram to show how they’d work this out, then you also provided some extending prompts. So can you give a quick run through of how that played out in the classroom? And did the students respond well to the extension prompts?

MM: Just to give you a bit of an idea about the narrative side of things – that task was based around a photo that I shared with the kids of my own family when we were on holidays playing chess on a chessboard. And I’ve had some really positive interactions with kids around that, where some kids will come and say to you ‘oh I play chess lots,’ and ‘I’m a big fan of chess’ so it’s building relationships there where they can say, you might share a common interest.

I mean I’ve had the other experience where I’ve been at a school and I’ve told the kids: ‘this giant chess board was at this particular holiday place in Queensland, and then I’ve had a student come back to me like two months later over the summer holidays and saying: ‘guess where I went over summer? We went and stayed at Paradise Resort and we played chess on that chessboard’ and the kid being really excited to share that with you.

So the narrative, that was the true part of the narrative. I mean the made-up part was – it talks about us having a round robin chess tournament. Now, we didn’t have a round robin chess tournament, we were actually there trying to enjoy, we didn’t spend all day at a giant chessboard playing chess.

So I think it’s teachers being able to feel comfortable taking parts of their life – you know, some real-world application – but also feeling free to be able to sort of elaborate, add to it, and make it work for the maths.

The task – yeah, it’s a really fantastic task because it’s got quite a low entry point, in that you could work on that task just sort of saying – you know it’s like the old problem where you say ‘there are eight people in a room and they all shake hands with each other. How many handshakes would there be?’ But it’s much more – I mean, the idea of playing chess against each other, students can visualise that a lot better and can sort of conceptualise it to say: ‘well if Nash plays against Isaiah, and then next Nash would play against Genevieve…’ so they can sort of work through all the combinations of who Nash would need to play.

Nearly every student you give that task to can enter the task and can have some level of success. But at the higher level it’s a very cognitively engaging task. I mean, the extension task is asking for them to basically find a formula of how to work out any triangular number. And so I used that task with Year 3/4 students and I’ve had students I’ve worked with in the Year 3/4 cohort who are able to sort of show you, ‘I can work out any triangular number and this is how you do it’. And they can show you visually how the formula works.

So I think that’s the beauty to this approach to teaching in that you’re really allowing for true differentiation. You’re presenting a task and there’s scope there for students to work there at a number of different levels.

DR: And as we mentioned very briefly before the classroom teachers were observing you first conducting these lessons before conducting the lessons themselves. Why was that important to do than just instructing teachers on how to run this and getting them to launch straight into it. Why was the observation element quite critical?

MM: I’m a big believer of if you want to get change happening within an organisation, it’s important to have buy-in from people. It’s important for people to actually believe that what you’re doing is going to be doing is beneficial. And for teachers, the vast majority of teachers, when they see that something is effective with their own students, you’ve won them over. So if they can see their own students being challenged in a way they previously haven’t been challenged.

I mean I had this experience yesterday when I was at a school in the western suburbs of Melbourne and I was working in a prep classroom and there was a prep student who, you know, traditionally didn’t really have a lot of success in the maths class and then this student produced some work and this classroom teacher was literally speechless. He was just blown away, he was like, ‘I cannot believe that he’s just done that, I’ve never seen him do that before’.

Now when I go back to work with that teacher in a fortnight’s time or whenever I’m back out there to work with them, they’re going to be much more receptive to this approach because they can see that it works.

And I think you’re also setting teachers up for success then. Because if they’ve seen that lesson structure a few times, the idea that it is very repetitive as a structure, it gives them something that they can sort of say, ‘right, now if I’m going to have a go at taking a lesson using this approach, these are the things that I want to do’. And it’s very easy to reproduce because they’ve seen it done a number of times.

So it’s both about supporting the teachers so they can have success, but also about generating that buy-in and I think that that comes – it’s one thing to deliver PD and to say ‘this is great’. It’s another thing for teachers to see it working with their own students.

DR: And so another big part of the study was how you actually measured the attitudes that students held towards these problem solving tasks and they were overwhelmingly positive. You’d mentioned before that this was kind of what you were expecting to happen because anecdotally you knew that students responded really well to these kinds of tasks. But something that I found really interesting was that they really enjoyed the challenging aspect of these problems and also the collaborative nature. So can you talk me through what the students said and wrote in their questionnaires about those two particular aspects?

MM: Yeah. I guess, I mean one thing that did surprise me was – I expected the results to be positive because that’s what I kind of see when I work not just at this school, but at lots of schools – I was surprised in the fact that of all the students that were involved in the study, that there was no one that expressed, like, negative attitude. Which, you know, was quite sort of gobsmacking for me.

But in terms of what they identified that made it enjoyable, engaging for them. Like you said, there was a couple things they touched on. So one was the idea of challenge. And I think this is something that sometimes teachers struggle with, this idea that: ‘if I make the work more challenging, the kids will disengage. They won’t persist, they won’t enjoy tackling the task’.

And I actually think that’s counter to everything we know about humans. If we think about ourselves as adults, if we’re given some sort of routine, mundane task to perform over and over again it’s every chance that we might do it if we have to do it, but we’re not going to enjoy it. But people love a challenge, people love being pushed cognitively and trying to see if they can be the first one to figure things out. I think humans love a challenge and if I enjoy a challenge as a 43-year-old, there’s no reason to think that like a six-year-old or a 12-year-old wouldn’t enjoy a challenge. So that’s come through to me anecdotally, you know, time and time again over the years, so it was good to see that come in through formally in the study that we did.

The other really big – and again, in some cases this really contrasts with the regular classroom practice – this idea of allowing the students to collaborate. And like I said before, not just allowing, but actively encouraging it. I think a lot of classroom teachers are concerned that if they let the kids move around the room and talk to each other, they’re going to lose control and it’s going to descend into chaos. But I think the two ideas that you’ve just asked about are connected. Like if they’re working on something that they think is worthwhile and challenging, they’re much more likely to stay on task.

And again, humans enjoy collaborating. Humans enjoy socialising, talking, sharing ideas. So if that’s the way, if I was to present PD [professional development] at a school and I was to do five hours of me talking and there’d be no opportunity for staff to actively engage and collaborate with each other, I mean, I would never be invited back to the school.

So then the question would be, well why do we get our students to do this? Why is a maths lesson me talking for 20 minutes telling you everything I know about place value, and then you working on a worksheet by yourself for half an hour and not being allowed to talk?

That’s not going to be enjoyable for us as adults. Why would it be enjoyable for an eight-year-old in a Year 2 class? So I think that in some cases the success that we have when we go and work in schools is partly because it’s such a sharp contrast to the regular practice in the school about the way maths is learned. And that if we can make mathematics more social, then we have much more chance of having students being engaged and wanting to learn.

DR: And is part of that as well – like you mentioned before – the fact that they have that five minutes at the beginning to concentrate on the problem as an individual and silently, but then they open up to the collaboration. Is that balance quite good and quite important?

MM: Yeah. It’s really crucial. And I always tell teachers that I’m working with that one’s not more important than the other, that they’re equally important. But if you let kids collaborate straight away then what you might find is that kids will just straight away – say you and I are working together, and you’re a stronger student in terms of your current performance in maths, well I might just be led by you, and you’ll just be telling me, ‘do this, do that’.

Much more likely if I’ve had some time to think and ponder on the task, that A, when I come to you, maybe I’ll have some questions about what I’m doing and you can guide me and direct me, rather than telling me what to do. But, B, there might be the chance that I may choose not to work with you, even thought we may be best friends, because I may see that someone else is approaching the problem with a similar mindset, a similar approach to me. Or I may choose to say in this instance: ‘I’m going to keep doing this by myself because I feel like I’m getting some momentum here. I can see that I’m making some progress’.

So I think that five minutes silent time is really crucial, and then it becomes really crucial (this becomes a classroom management thing) as a teacher, you have to be able to make sure that it is truly five minutes and it is truly silent and it is truly independent and also truly productive. Because it’s no good them just sitting silently looking at the clock, you know, looking at the stop watch counting down before they bang go into talking to each other.

So the way you know that’s productive is when you see the kids are on task. When you see the tops of their heads looking down at their page, and they’re thinking, and they’re gathering materials. And you can tell really clearly as a teacher when that’s not happening.

DR: And so just finally then, I’m thinking now for teachers who are listening to this episode who are thinking they want to implement a similar approach in their maths classroom for students of a similar age, is there anything that we haven’t covered already that would be good to keep in mind? Or perhaps some good first steps to take?

MM: Yeah, look I think that the model that I see that works really well is I think what we spoke about before. Is that you have to have people that are able to model what it should look like to be able to win teachers over, for them to be able to say ‘I can see the benefit of this, I can see how this works’. So whether that be – I mean, I’m definitely not trying to spruik for work – but whether that be internally – you know, like a lot of schools have really great classroom teachers. Some of those people have moved into learning specialist roles.

But whether that be internally with those people, like give them the time to go into other people’s classrooms and to be able to model this type of approach and to show the classroom teachers how it works and to be able to answer those questions. Or whether it be externally, by bringing in consultants who have the skill and expertise to do it, I think that’s really important. I think it’s important that people see it in practice first before they try to do it.

And it’s also really important, as well as seeing lessons, that people have time to then unpack the lesson and talk about it together. So if you’ve got a learning specialist at your school that’s modelling this type of lesson for, say a graduate teacher, there needs to be some time allocated for them to sit. Because the graduate teacher may walk away saying, ‘that was a great lesson’. But the next step is them being able to identify why was it a great lesson? What worked? And what can they do to plan a similar great lesson the following week?

Because, you know, if you just say ‘well, that was a great lesson, but I can’t do that lesson again because my kids have already done it, so where do I go with it?’ Whereas if you can identify and say: ‘oh I see what worked well. The thing that worked well is they were engaged with the problem.’ Why were they engaged in the problem? ‘It had a real world link’. Why was your questioning effective during the lesson? ‘Well, it was because you knew, you had a clear focus of what the content was’. What are we focusing here? What’s the mathematical concepts we’re focusing on?

So, as a classroom teacher you know the right question to ask and the right student at the right time and there’s a lot of work that goes into that, but like I said, it’s definitely something that’s attainable for all classroom teachers with the right support.

That’s all for this episode. Thanks for listening. Be sure to subscribe to our podcast channel on Spotify , Apple podcasts or SoundCloud , so you can be notified of any new episodes. While you're there, we'd love for you to rate and review the podcast in your podcast app.

Russo, J., & Minas, M. (2020). Student Attitudes Towards Learning Mathematics Through Challenging, Problem Solving Tasks: “It’s so Hard– in a Good Way”. International Electronic Journal of Elementary Education, 13 (2), 215-225. https://doi.org/10.26822/iejee.2021.185 .

Michael Minas says he believes sometimes teachers struggle with the idea that: ‘if I make the work more challenging, the kids will disengage. They won’t persist, they won’t enjoy tackling the task’.

Reflect on a recent lesson you taught. How challenged would you say students were? How do you know the level of challenge was appropriate? Do you think you could have challenged students further? Were there opportunities for students to participate in extension tasks?

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What is Subitising? A hidden gem in developing mental arithmetic skills

When they practise subitising, children develop their ability to recognize and quickly comprehend the quantities of objects, a skill that can help them with mental calculations, estimation and problem-solving.

In a world where calculators and smartphones have made mental arithmetic seem less important, it's crucial to remember the value of building strong foundational maths skills in children. One often-overlooked method that can greatly improve mental arithmetic is subitising. In this blog post, we'll explore the concept of subitising, its connection to mental arithmetic and how you can help your child develop these essential skills.

Subitising plays an important role in how young children learn and is a key part within the requirements of the National Curriculum. It is formally introduced in the Early Years Foundation Stage ( EYFS ), where children begin to develop this skill as part of learning number sense. By understanding and nurturing subitising skills, educators and parents can lay a strong foundation for a child's mathematical journey.

Understanding subitising

When they practise subitising, children develop their ability to recognise and quickly comprehend the quantities of objects through number patterns, a skill that can help them with mental calculations, estimation and problem-solving.

What is subitising?

Subitising — from the Latin word meaning ‘suddenly’ — is the skill of instantly seeing how many items are in a group . For example, if I hold up my hand, you normally wouldn’t have to count my fingers and thumb to know it’s five. Most people can subitise as many as six or seven randomly arranged items.

In the EYFS, children typically start by subitising up to 5, gradually progressing to understanding numbers up to 10. This progression aligns with the development of their overall number sense and prepares them for more advanced mathematical concepts in later years.

Editor’s note: There are some exceptions. Read our ‘ parents and teachers guide to identifying specific learning difficulties ’ to learn more.

Difference between perceptual and conceptual subitising

There are two primary types of subitising: perceptual and conceptual. According to the website Early Math Counts , perceptual subitising is instantly knowing how many are in a given set, while conceptual subitising is the ability to see sets of numbers within larger sets, such as seeing two fours in the eight of a domino. If the number of items is too large to see all at once our brains ‘decompose’ the group into smaller chunks before adding them back together.

Here are some examples to illustrate these types:

• Perceptual Subitising : Quickly recognizing that there are four dots on a dice without counting
• Conceptual Subitising : Seeing a domino with 6 dots and instantly recognizing it as two groups of three
• Advanced Conceptual Subitising : Looking at a 10-frame with 7 counters and immediately seeing it as a full five and two more

The role of subitising in early years maths education

Sue Gifford at the University of Roehampton explains that subitising helps children build images for numbers, visualise and learn number facts . Very young children are usually able to recognise the number of items in a group without having to count them. For instance, most four-year-olds recognise five dots on a dice, which helps them understand the cardinal value — or 'howmanyness' — of five. They can then link to the word ‘five’ and the numeric symbol ‘5.’

The Connection Between Subitising and Mental Arithmetic

How subitising develops number sense.

Subitising is important because it helps children gain a deep understanding of quantity. Without an understanding of quantity, they may have no option but to count everything when they start to face more complex problems.

John Van de Walle, a prominent mathematics educator and author , emphasised the importance of subitising in developing number sense and mathematical fluency in young children. He said subitising lays the foundation for more advanced mathematical understanding by allowing children to grasp the concept of number as a quantity, rather than just a symbol or a sequence. By being able to recognize small groups of objects without counting, children can develop a strong sense of numbers and their relationships.

As children progress through Key Stage 2, their ability to subitise can significantly enhance their understanding and application of column addition and subtraction.

For example:

• Quick recognition of number bonds within 10 (developed through subitising) can speed up mental calculations in column addition.
• Understanding of number composition (e.g., seeing 8 as 5 and 3) helps in regrouping during written subtraction.
• The ability to visualise quantities supports estimation skills, which are crucial for checking the reasonableness of answers in formal calculations.

The role of subitising in fostering mental arithmetic skills

There are several ways in which the skill of subitising contributes to the development of mental arithmetic abilities:

• Rapid recognition of small quantities : by instantly recognising quantities, children can perform mental calculations more rapidly, which leads to improved mental arithmetic skills
• Number combinations : subitising helps children develop an understanding of number combinations or number bonds. For example, when children are presented with the dots on a pair of dice, subitising allows them to see the numbers as parts that make up a whole, which supports mental arithmetic skills such as addition and subtraction
• Number decomposition : subitising helps in number decomposition, or breaking a number down into its component parts. This allows children to quickly recognise different smaller groups within the larger set, which is crucial for mental arithmetic skills such as multiplication
• Mental estimation : by quickly perceiving quantities without counting, children can estimate the total number of objects or the result of an operation more effectively, which is a valuable skill in various mental arithmetic tasks that require approximations and quick calculations
• Building number sense : subitising supports the development of an intuitive understanding of numbers and their relationships

The link between subitising and other cognitive abilities

A strong understanding of numbers continues to develop over time through the exploration of numbers, visualising numbers and relating to numbers in different ways. Judy Hornigold provides this checklist for tracking how children are progressing with number sense :

• An awareness of the relationship between number and quantity
• An understanding of number symbols, vocabulary, and meaning
• The ability to engage in systematic counting , including notions of cardinality — knowing the symbol 4 denotes a group of four objects — and ordinality — knowing that 4 comes before 5 and after 3.
• An awareness of magnitude, or the ability to compare groups
• An understanding of different representations of number
• Competence with simple mathematical operations

Subitising is also linked to the following cognitive skills:

• Visual perception : the ability to discriminate and differentiate small groups of objects quickly
• Working memory : the ability to hold a mental representation of the quantity and rapidly process it
• Attention : the ability to focus attention on the visual stimuli and selectively attend to relevant information
• Numerical cognition : understanding and manipulating numerical quantities
• Executive functioning : a capacity for mental flexibility, inhibition of irrelevant information and cognitive control, which contribute to the development of executive functions involved in goal-directed behaviour, self-regulation, and problem-solving, since children must quickly process the quantity and inhibit the impulse to count

Activities to Enhance Subitising and Mental Arithmetic Skills

There are a variety of ways to practise, but one fun way is through introducing subitising games in the classroom. Let’s take a look at a few that support early years learning.

Dot card games

Dot card games present a variety of dot arrangements that help children develop their mental image of specific quantities. Try this downloadable dot card game from Young Mathematicians called ‘ Which One Doesn’t Belong .’ As children practise, explain how chunking dots into groups can make it easier to subitise. For example, chunking two dots and two dots helps you see that there are four dots.

For a deeper dive into dot patterns and cards using early number sense, see this article by Jennie Back .

Ten-frame activities

Ten-frames not only help children learn to subitise, but also to develop their understanding of ten as one of the most significant numbers in our number system. The National Council of Teachers of Mathematics has created a cool applet that helps develop addition and counting skills through playing games with ten-frames.

These informative blogs take a closer look at ten-frames and number sense:

• ‘ Early years maths mastery: introducing the CPA approach ,’ by  Sabrina Pinnock
• ‘ Number Sense Series: A Sense of 'Ten' and Place Value ,’ by Jenni Way

An example of ten frames built inside the Maths — No Problem! Visualiser app

Number line games

Teachers have used number lines for decades to teach number sense and arithmetic. Try these number line activities to help children strengthen their subitising skills:

• Use sticky notes to line up numbers
• Build a Lego number line
• Create a life-size number line (see video below)

Dice games are useful for helping children build images for numbers, visualise and learn number facts. Here are some good dice games from The Teacher Next Door , including ‘Build it Big’ and ‘Roll and Round.’ We Are Teachers also has some fun ideas for dice games including ‘Tenzi’ and ‘Dice Wars.’

Using hands and fingers in games

Hold up a number of fingers quickly and ask children to say how many they see. Gradually increase the difficulty by using both hands or asking children to show numbers in different ways.

You can use this video to quickly set up the exercise as you teach subitising.

Tips for Parents and Educators

Encourage number talks and discussions.

Talking with their peers, asking questions, and debating best approaches to problem-solving help students develop more complex thinking and reasoning skills, says Nina Parrish in Edutopia . Psychologists such as Jean Piaget and Lev Vygotsky also advocated the idea of metacognitive talk because children benefit from being active participants in the construction of knowledge. Conversations can create productive conflict that helps students develop multiple perspectives, leading to deeper learning.

There are many sources for exploring maths talks:

• Catherine Reed on her website The Brown Bag Teacher offers some unique methods for encouraging number talks
• The National Association for the Education of Young Children gives several tips for supporting maths readiness through maths talks
• Ruth Parker’s Number talks are a great way to develop number flexibility with your children or pupils
• Here’s Jo Boaler from Stanford University teaching a visual dot card number talk .

In addition, there are several strategies for encouraging metacognitive talk:

• Limit teacher talk time: most of the thinking and reasoning work should be left to the students
• Use open-ended questions and encourage problem-solving: try to ask more questions that are open-ended and encourage students to find the answer on their own or explore the process to find the answer collaboratively
• Explain the steps or outline the process: encourage students to explain their thinking process and how to arrive at an answer. Journalling is an excellent way to do this
• Generate knowledge and new examples: have students create their own unique examples and then collaborate with a partner to compare and contrast what they came up with and check each other’s work
• Take on a specific role in the critical thinking process: talk can be used as a scaffold that allows students to engage in assignments with increased rigour. Suggest strategies such as reciprocal teaching , where students work in groups to analyse a complex fiction or nonfiction text or sort through a maths word problem, each taking on a specific thinking role in order to practise making predictions, asking questions, clarifying and summarising

Incorporating subitising activities into daily routines

There are many ways to blend subitising activities into your daily routine. Some examples of these for the classroom from Rhody Girl Resource are brain breaks and time fillers. For some tips on how to encourage early maths skills at home, check out this blog post from The Preschool Toolbox , which advises parents to use mittens, socks, play dough pieces and dice to create small groups of items. Once your child can count a group correctly, progress to visual scanning for mental maths. Learning how to do quick visual scans will help your children strengthen their subitising skills.

From Jump Math , here are some examples of how to develop numeracy skills at home, including ‘The Bedtime Maths Challenge’ and ‘Bathroom Tiles.’

Creating a supportive learning environment

“The single most important factor in a child’s schooling is how they feel about themselves as individuals and as learners within the classroom,” says Katie Bowles, who looks at ‘ the psychologically safe classroom and restorative practice ' through the work of Becky Carlzon, co-author of The Learning Powered Approach and Mark Finnis, who wrote Restorative Practice , in which he explores the value of relationships within a school setting.

According to Teresa Duncan in Math for All , the saying ‘Maslow before Bloom’ sums up the decades of research that show how a sense of physical and emotional safety is the foundation for the development of social, emotional, and academic competencies. In other words, safe, supportive, classroom learning environments are important in supporting mathematics achievement.

This is demonstrated in a study by Rebekah Berlin and Julie Cohen , in which they analyse 400 lessons from 49 elementary schools. They discovered that the highest degree of mathematical engagement was found only when the learning environment was emotionally supportive.

To emphasise this point, a 2022 study in Frontiers of Psychology found that among several factors, “classroom climate provides the most critical microsystems directly affecting student learning process and outcomes.” The study further explains classroom climate includes at least three essential components — teacher-student interaction, instructional support and social-emotional support.

By understanding the concept of subitising and incorporating it into your child's learning, you can help them build a strong foundation in maths. With consistent practice and encouragement, your child will soon be able to make quick mental calculations, setting them up for success by helping them build maths skills for the future.

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Title: mathscape: evaluating mllms in multimodal math scenarios through a hierarchical benchmark.

Abstract: With the development of Multimodal Large Language Models (MLLMs), the evaluation of multimodal models in the context of mathematical problems has become a valuable research field. Multimodal visual-textual mathematical reasoning serves as a critical indicator for evaluating the comprehension and complex multi-step quantitative reasoning abilities of MLLMs. However, previous multimodal math benchmarks have not sufficiently integrated visual and textual information. To address this gap, we proposed MathScape, a new benchmark that emphasizes the understanding and application of combined visual and textual information. MathScape is designed to evaluate photo-based math problem scenarios, assessing the theoretical understanding and application ability of MLLMs through a categorical hierarchical approach. We conduct a multi-dimensional evaluation on 11 advanced MLLMs, revealing that our benchmark is challenging even for the most sophisticated models. By analyzing the evaluation results, we identify the limitations of MLLMs, offering valuable insights for enhancing model performance.

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