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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.

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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:

The problem	How to act out the problem
Gia has 6 apples. Jordan has 3 apples. How many apples do they have altogether?	Two students use counters to represent the apples. One student has 6 counters and the other student takes 3. Then, they can combine their “apples” and count the total.
Michael has 7 pencils. He gives 2 pencils to Sarah. How many pencils does Michael have now?	One student (“Michael”) holds 7 pencils, the other (“Sarah”) holds 2 pencils. The student playing Michael gives 2 pencils to the student playing Sarah. Then the students count how many pencils Michael is left holding.

3. Work backwards

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

For example,

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

4. Write a number sentence

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

5. Use a formula

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

Strategies for checking the solution 

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

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

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

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

2. Estimate to check for reasonableness

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

3. Plug-In Method

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

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

4. Peer Review

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

5. Use a Calculator

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

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

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

Polya’s 4 steps include:

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

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

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

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

How Third Space Learning improves problem-solving 

Resources .

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

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

One-on-one tutoring 

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

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

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

Problem-solving

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

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

READ MORE :

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

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

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

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

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

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Rich Problems – Part 1

Rich problems – part 1, by marvin cohen and karen rothschild.

One of the underlying beliefs that guides Math for All is that in order to learn mathematics well, students must engage with rich problems. Rich problems allow ALL students, with a variety of neurodevelopmental strengths and challenges, to engage in mathematical reasoning and become flexible and creative thinkers about mathematical ideas. In this Math for All Updates, we review what rich problems are, why they are important, and where to find some ready to use. In a later Math for All Updates we will discuss how to create your own rich problems customized for your curriculum.

What are Rich Problems?

At Math for All, we believe that all rich problems provide:

• opportunities to engage the problem solver in thinking about mathematical ideas in a variety of non-routine ways.
• an appropriate level of productive struggle.
• an opportunity for students to communicate their thinking about mathematical ideas.

Rich problems increase both the problem solver’s reasoning skills and the depth of their mathematical understanding. Rich problems are rich because they are not amenable to the application of a known algorithm, but require non-routine use of the student’s knowledge, skills, and ingenuity. They usually offer multiple entry pathways and methods of representation. This provides students with diverse abilities and challenges the opportunity to create solution strategies that leverage their particular strengths.

Rich problems usually have one or more of the following characteristics:

• Several correct answers. For example, “Find four numbers whose sum is 20.”
• A single answer but with many pathways to a solution. For example, “There are 10 animals in the barnyard, some chickens, some pigs. Altogether there are 24 legs. How many of the animals are chickens and how many are pigs?”
• A level of complexity that may require an entire class period or more to solve.
• An opportunity to look for patterns and make connections to previous problems, other students’ strategies, and other areas of mathematics. For example, see the staircase problem below.
• A “low floor and high ceiling,” meaning both that all your students will be able to engage with the mathematics of the problem in some way, and that the problem has sufficient complexity to challenge all your students. NRICH summarizes this approach as “everyone can get started, and everyone can get stuck” (2013). For example, a problem could have a variety of questions related to the following sequence, such as: How many squares are in the next staircase? How many in the 20th staircase? What is the rule for finding the number of squares in any staircase?

• An expectation that the student be able to communicate their ideas and defend their approach.
• An opportunity for students to choose from a range of tools and strategies to solve the problem based on their own neurodevelopmental strengths.
• An opportunity to learn some new mathematics (a mathematical residue) through working on the problem.
• An opportunity to practice routine skills in the service of engaging with a complex problem.
• An opportunity for a teacher to deepen their understanding of their students as learners and to build new lessons based on what students know, their developmental level, and their neurodevelopmental strengths and challenges.

Why Rich Problems?

All adults need mathematical understanding to solve problems in their daily lives. Most adults use calculators and computers to perform routine computation beyond what they can do mentally. They must, however, understand enough mathematics to know what to enter into the machines and how to evaluate what comes out. Our personal financial situations are deeply affected by our understanding of pricing schemes for the things we buy, the mortgages we hold, and fees we pay. As citizens, understanding mathematics can help us evaluate government policies, understand political polls, and make decisions. Building and designing our homes, and scaling up recipes for crowds also require math. Now especially, mathematical understanding is crucial for making sense of policies related to the pandemic. Decisions about shutdowns, medical treatments, and vaccines are all grounded in mathematics. For all these reasons, it is important students develop their capacities to reason about mathematics. Research has demonstrated that experience with rich problems improves children’s mathematical reasoning (Hattie, Fisher, & Frey, 2017).

Where to Find Rich Problems

Several types of rich problems are available online, ready to use or adapt. The sites below are some of many places where rich problems can be found:

• Which One Doesn’t Belong – These problems consist of squares divided into 4 quadrants with numbers, shapes, or graphs. In every problem there is at least one way that each of the quadrants “doesn’t belong.” Thus, any quadrant can be argued to be different from the others.
• “Open Middle” Problems – These are problems with a single answer but with many ways to reach the answer. They are organized by both topic and grade level.
• NRICH Maths – This is a multifaceted site from the University of Cambridge in Great Britain. It has both articles and ready-made problems. The site includes  problems for grades 1–5 (scroll down to the “Collections” section) and problems for younger children . We encourage you to explore NRICH more fully as well. There are many informative articles and discussions on the site.
• Rich tasks from Virginia – These are tasks published by the Virginia Department of education. They come with complete lesson plans as well as example anticipated student responses.
• Rich tasks from Georgia – This site contains a complete framework of tasks designed to address all standards at all grades. They include 3-Act Tasks , YouCubed Tasks , and many other tasks that are open ended or feature an open middle approach.

The problems can be used “as is” or adapted to the specific neurodevelopmental strengths and challenges of your students. Carefully adapted, they can engage ALL your students in thinking about mathematical ideas in a variety of ways, thereby not only increasing their skills but also their abilities to think flexibly and deeply.

Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning. Thousand Oaks, CA: Corwin Mathematics.

NRICH Team. (2013). Low Threshold High Ceiling – an Introduction . Cambridge University, United Kingdom: NRICH Maths.

The contents of this blog post were developed under a grant from the Department of Education. However, those contents do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government.

Math for All is a professional development program that brings general and special education teachers together to enhance their skills in planning and adapting mathematics lessons to ensure that all students achieve high-quality learning outcomes in mathematics.

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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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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

• Inside Mathematics
• Math Pathways (DCMP)
• Keywords Search
• inside problem solving

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.

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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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. 

Do you love math hands-on activities? Join our weekly newsletter and get the  F REE Fractions Cut and Paste Worksheet that is perfect for back-to-school and Halloween! 

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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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Open-Ended Tasks and Questions in Mathematics

by CristinaM. | Sep 13, 2014 | inquiry , math , thinking | 5 comments

One way to differentiate in math class is creating open-ended tasks and questions (I talked about several differentiation strategies I use here – Mathematically Speaking ).

I think it is useful to clarify the scheme of mathematical problems – below I used Foong Pui’s research paper:

“Problems in this classification scheme have their different roles in mathematics instruction as in teaching for problem solving, teaching about problem solving, or teaching via problem solving.”

1.   CLOSED problems   are well-structured problems in terms of clearly formulated tasks where the one correct answer can always be determined in some fixed ways from the necessary data given in the problem situation.

 A. Routine closed problems – are usually multi-step challenging problems that require the use of a specific procedure to arrive to the correct, unique, answer.

B. Non-routine closed problems – imply the use of heuristics strategies * in order to determine, again, a single correct answer.

*Problem-solving heuristics: work systematically, tabulate the data, try simpler examples, look for a pattern, generalize a rule etc.

Routine problem : Minah had a bag of rice. Her family ate an equal amount of rice each day. After 3 days, she had 1/3 of the rice left. After another 7 days, she had 24 kg of rice left. How much rice was in the bag at first?

Non-routine problem : How many squares are there in a chess board?

2. OPEN –ENDED problems – are often named “ill-structured” problems as they involve a higher degree of ambiguity and may allow for several correct solutions. Real-life mathematical problems or mathematical investigations are of this type – e.g. “How much water can our school save on a period of four months?” or “Design a better gym room considering the amount of money we can spend.”

FEATURES of open-ended problems :

• There is no fixed answer (many possible answers)
• Solved in different ways and on different levels (accessible to mixed abilities)
• Empower students to make their own mathematical decisions and make room for own mathematical thinking
• Develop reasoning and communication skills

HOW do you create open-ended tasks?

Usually, in order to create open-ended questions or problems, the teacher has to work backwards :

• Indentify a mathematical topic or concept.
• Think of a closed question and write down the answer.
• Make up a new question that includes (or addresses) the answer.

STRATEGIES to convert closed problems/questions

• Turning around a question

CLOSED: What is half of 20?

OPEN: 10 is the fraction of a number. What could the fraction and the number be? Explain.

CLOSED:  Find the difference between 23 and 7.

OPEN: The difference between two numbers is 16. What might the numbers be? Explain your thinking.

CLOSED: Round this decimal to the decimal place 5.7347

OPEN: A number has been rounded to 5.8. What might the number be?

CLOSED: There are 12 apples on the table and some in a basket. In all there are 50 apples. How many apples are in the basket?

OPEN: There are some apples on the table and some in a basket. In all there are 50 apples. How many apples might be on the table? Explain your thinking.

• Asking for similarities and differences.

Choose two numbers, shapes, graphs, probabilities, measurements etc. and ask students how they are alike and how they are different.

Example: How are 95 and 100 alike? How are they different?

Possible answers:

They are alike because you can skip count by 5s, both are less than 200, both are greater than 90 etc.

They are different because one is a three-digit number, only one ends in 5, only one is greater than 99 etc.

Example: How are the numbers 6.001 and 1.006 alike? How are they different?

• Asking for explanations.

Example: Compare two fractions with different denominators. Tell how you compare them.

Example: 4 is a factor for two different numbers. What else might be true about both numbers?

• Creating a sentence

Students are asked to create a mathematical sentence that includes certain numbers and words.

Example: Create a sentence that includes numbers 3 and 4 along with the words “more” and “and”.

• 3 and 4 are more than 2
• 3 and 4 together are more than 6
• 34 and 26 are more than 34 and 20 etc.

Example: Create a question involving multiplication or division of decimals where the digits 4, 9, and 2 appear somewhere.

Example: Create a sentence involving ½  and 64 and the words “less” and “twice as much”.

• Using “soft” words.

Using the word “close” (or other equivalents) allows for a richer, more interesting mathematical discussion.

Example: You multiply two numbers and the product is almost 600. What could the numbers have been? Explain.

Example: Add two numbers whose sum is close to 750. What can the numbers be? Explain.

Example: Create two triangles with different but close areas. (*instead of, “Create a triangle with an area of 20 square inches.”)

……………………………………………………………………………………………………………………………………………………………………………………………………

A few important considerations are to be made when creating open-ended problems or questions.

• Know your mathematical focus .
• Develop questions with the right degree of ambiguity (vague enough to be interesting and to allow for different responses, but not too vague so as students get frustrated).
• Plan for two types of prompts :
• enabling prompts (for students who seem unable to start working)
• extension prompts (for students who finish quickly)

High quality responses from students have the following features:

• Are systematic (e.g. may record responses in a table or pattern).
• If the solutions are finite, all solutions are found.
• If patterns can be found, then they are evident in the response.
• Where a student has challenged themselves and shown complex examples which satisfy the constraints.
• Make connections to other content areas.

……………………………………………………………………………………………………………………………………………………………………………………………………………….

References:

Designing Quality Open-Ended Tasks in Mathematics , Louise Hodgson, 2012

Using Short Open-ended Mathematics Questions to Promote Thinking and Understanding , Foong Pui Yee, National Institute of Education, Singapore

Good Questions – Great Ways to Differentiate Mathematics Instruction , Marian Small, 2012

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Thank you for posting this. I appreciate how there is a comparison between the two (closed and open ended) types of questions and the considerations that go along with each. Thanks!

You are welcome!

Wow, well-written, thank you. I’m excited that my teaching is getting great, clear, and organized at the level that I’m at. But this article reminds me there are many higher levels I can get to, including this area of more open-endedness. Thank you!

I am happy to have helped even in a small way!

Thank you so much

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25+ Engaging Math Tasks That Promote a Growth Mindset

In our culture, we are bombarded with messages implying that some people are good at math and some people aren’t. There’s this notion that some people have that elusive “math gene” and some people don’t. Overcoming these negative attitudes and baggage to encourage kids that they can in fact learn and enjoy math and there is actually no such thing as a “math person,” is a challenge. But more and more research is showing up that kids with a growth mindset towards math do better on standardized tests, are more engaged in class and have a better attitude about math in general. Today I want to focus on how rich math tasks can excite and engage kids and begin to develop a growth mindset .  As you go through this list of math tasks that promote a growth mindset , pick out your favorites to try with your kids!

* Please Note : This post contains affiliate links which help support the work of this site. Read our full disclosure here .*

What is a Growth Mindset?

The notion of a fixed mindset versus a growth mindset began with the work of Carol Dweck . She explains that everyone has a mindset or idea about how they learn. Those with a fixed mindset believe that you can’t change your level of intelligence. You can only learn so much or go so far.

Those with a growth mindset, on the other hand, view intelligence as something that can be achieved through hard work and perseverance .

In other words, “smarts” is not something you’re born with, it’s something that can develop over time if you’re willing to put in the effort.

Because so many in our culture have a fixed mindset towards math, it is important that we do what we can to dispel these notions as more and more brain research sheds light on how we learn new things. Because the truth is that our brains grow when we make mistakes . And our brains can grow and change and adapt even if half of the brain is removed .

If the human brain is capable of these remarkable achievements, surely our brains are capable of learning Algebra !

How to Develop a Growth Mindset Towards Math:

Changing our own mindset towards learning math can be a challenge, much less changing our kids’ mindsets . But it’s not impossible.

One thing that has to change is the way math is viewed . All too often, math is seen as a closed, fixed subject to be memorized and then forgotten.

But math is a creative, open and exciting topic that touches every single aspect of our lives! We see and use math everyday, and to help kids see that they can learn and achieve math at high levels, we have to get them excited about it.

One way to encourage kids to see math as creative and present in the world around us is to explore math in nature . These concepts are great because they are accessible for young kids but complex enough to challenge older, advanced learners .

Jo Boaler, in her book Mathematical Mindsets , explains the importance of rich and open math tasks. No matter what curriculum you use, you are the teacher and have the opportunity to present math in a way that is engaging.

In the book, Boaler discusses several ways tasks can be rich and engaging for kids . Here is a quick overview:

• Can you open the task to encourage multiple methods, pathways and representations?
• Can you make it an inquiry task?
• Can you ask the problem before teaching the method?
• Can you add a visual component?
• Can you make it low floor and high ceiling?
• Can you add the requirement to convince & reason?

Using these criteria, I want to share a growing list of math tasks and websites where you can find meaningful lessons to engage your kids in math learning and teach with a growth mindset .

And if you have any ideas to add to this list, be sure to share in the comments!

Math Tasks That Promote a Growth Mindset:

Which Number Doesn’t Belong? I love this set of challenges because there are countless answers for each set of numbers. This is sure to spark some interesting math debates and deep thinking! (all ages)

How Many Seeds in a Pumpkin? This fun, hands on lesson is perfect for fall and can be combined with a fun math read aloud as well. (grades K-2)

Addition & Subtraction Equation Search : These fun challenges are more than just addition and subtraction and there are lots of different right answers. (grades 1-2)

Geoboard Challenge Cards : This set of challenges promotes exploration and discovery and includes some basic shapes and more advanced challenges. (all ages)

Compose Shapes with Geometiles : These challenging visual puzzles are created to use with Geometiles and are a great visual challenge for kids. (all ages)

What’s for Lunch? A Real Life Decimal Lesson : This introduction to adding and subtracting decimals is a great example of math in the real world. (grades 3+)

Gumball Estimation : This printable gumball challenge has different variations for increasing challenges or different ages. (grades 3+)

Surface Area of 3D Shapes : I have two investigations for determining the surface area of 3D shapes. Surface area of prisms and cylinders | Surface area of pyramids and cones (grades 7+)

Pattern Block Fraction Games : Pattern blocks provide a fantastic visual for understanding fractions. These games provide practice for kids to help them understand difficult operations.(grades 3-5):

Subtracting Mixed Numbers Game | Adding Fractions Game

Cut & Paste Logic Puzzles : Similar to sudoku puzzles, these are a fun introduction for young kids. (grades PreK-K)

Pattern Block Logic Puzzles : These puzzles are similar to sudoku, but are perfect for young kids not yet ready for more advanced logic puzzles. (grades k-2)

Grid & Sudoku Logic Puzzles : Older kids will love the challenge of these unique math problems that we don’t often see in textbooks. (grades 4+):

Thanksgiving Puzzles | Christmas Puzzles | Valentine’s Day Puzzles | 4th of July Puzzles

Missing Number Puzzles: Addition & Subtraction : These digital puzzles challenge kids to go beyond the standard algorithm for adding & subtracting large numbers and encourages deep thinking. (grades 3-4)

Pattern Puzzles for Google Slides : This set of missing number challenges encourages kids to see patterns, while building algebraic thinking. (grades 4-6)

Order of Operations Puzzles for Google Slides : Each problem in this set includes an expression with some of the numbers missing. Students must then use their knowledge of order of operations to figure out which number is missing.

Christmas Equations Algebra Challenge : These engaging problems help kids work on their algebraic reasoning and problem solving as more of a visual puzzle than a math problem. (grades 4+)

Exploring the Angles in Triangles : This hands on geometry challenge will show kids important facts about the relationships between angles in a triangle. (grades 6+)

What Makes a Triangle a Triangle? Use this lesson along with the book, “The Greedy Triangle” or on it’s on to discover what really makes a triangle a triangle. (all ages)

Would You Rather…? Tasks for Google Slides : This unique set of challenges covers ratios & percents by giving students a scenario and then asking a “would you rather…?” question. This forces them to think about the situation and the implications of the math involved.

Which Cup Holds the Most Hot Chocolate? This challenge teaches kids to think about volume as they compare different size cups. (grades 7+)

Estimating the Area of a Circle : This fun geometry challenge will help kids estimate and think about how to find the area inside a circle. (grades 7+)

Jumping Maze : This fun challenge can be done with sidewalk chalk or with pencil and paper.

Decorate a Christmas Tree : This open ended challenge is great for all ages! It can be simple enough for Kindergarten or challenging enough for high school. (all ages)

Look for Patterns in Pascal’s Triangle : This set includes several different patterns to color and observe in Pascal’s Triangle. (grades 2+)

Analyze Math Mistakes : Help kids learn from their mistakes and see them as opportunities for brain growth with these templates & classroom posters.

>> Buy Error Analysis Here!

Order of Operations Error Analysis : Deepen an understanding of order of operations with this set of order of operations error analysis tasks.

Math Websites with More Meaningful Math Tasks:

YouCubed : This site is run by Jo Boaler and her team at Stanford and includes a variety of tasks for kids of all ages.

You can also find tasks that are part of the “Week of Inspirational Math” here .

NCTM Illuminations : This has great math tasks that you can search by grade level or math standard.

GeoGebra : Find math activities and resources for all ages using their online tools for exploration.

NRich : This site offers lots of support for teachers as well as fun problems open for solution. You can search for problems to try based on age, and then you can even submit your solutions to them! The best solutions get shared on their site.

Estimation 180 : Find fun and engaging estimation challenge to get your kids thinking and problem solving.

Visual Patterns : This site offers a huge assortment of visual patterns for kids to explore and extend. This was one of my favorite ways to teach new concepts when teaching Algebra 1. This site would have made it so much easier for me!

Number Strings : Number strings are sets of related math problems, designed to support students to construct big ideas about mathematics and build their own strategies. These sets of problems can be done in small groups or as a whole class.

Yummy Math : This is a great resource for finding real world math problems that you can present to students before they’ve actually learned a method.

Mindset Mathematics Curriculum Series:

Finally, I want to share some curriculum supplement resources from Jo Boaler and the YouCubed team. Here are the books available so far, with more coming out soon:

• Mindset Mathematics | Grade 3
• Mindset Mathematics | Grade 4
• Mindset Mathematics | Grade 5
• Mindset Mathematics | Grade 6
• Mindset Mathematics | Grade 7

So that is my huge list of rich math tasks that promote a growth mindset!

I know it might be overwhelming at first to go through all these resources, so try to pick just one or two tasks to try for now , and then continue to weave them into your math routine as you get comfortable.

A lot of these open ended types of explorations are very different from how math has traditionally been taught, so it can be an adjustment to shift how you teach and think about math.

So just take it one day and one task at a time and have fun learning alongside your kids!

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Building Problem-Solvers: Engaging Maths Challenges & Playful Games for Primary Students

Table of Contents

Building Problem-Solvers: Incorporating challenging maths puzzles and games into primary education is a powerful strategy for developing problem-solving skills in young learners. We recognise the importance of engaging pupils with interactive tasks that not only stimulate their mathematical abilities but also build their confidence in tackling complex challenges. By designing diverse mathematical exercises that range from simple to complex, we create opportunities for children to develop fluency as mathematical thinkers and problem solvers, a skill set that is essential in today’s fast-paced, ever-evolving world.

It’s crucial to foster a supportive maths culture within the classroom, one that encourages risk-taking and persistence in overcoming obstacles. Education in primary mathematics should not only focus on procedural fluency but also on the understanding of fundamental mathematical concepts . Tools and strategies such as visual representations and heuristic approaches can be invaluable in promoting this kind of deep comprehension. By integrating these techniques with a varied set of maths puzzles and games, we make learning mathematics an engaging and dynamic experience.

Key Takeaways

• Developing problem-solving skills in maths is essential for primary education.
• Diverse and challenging puzzles enhance mathematical thinking.
• A supportive learning environment encourages perseverance and growth in maths.

Understanding the Fundamentals of Maths

In our pursuit of developing proficient problem solvers, it’s essential to start with a strong foundation in mathematics. We focus on building an understanding of numbers and operations, as well as recognising the significance of patterns and relationships. These components are vital in nurturing a comprehensive mathematical mindset.

Number and Operations

At the core of mathematical understanding are numbers and operations . This involves grasping how numbers work and interact with one another through basic operations : addition, subtraction, multiplication, and division. For example:

Addition: It’s combining quantities to increase the total.

Subtraction: It’s determining how much one quantity is greater than another

Multiplication : It simplifies repeated addition.

Division: It’s finding out how many times a number can be subtracted from another in equal parts.

Understanding these operations enables children to solve real-world problems effectively.

Patterns and Relationships

Patterns and relationships form the building blocks for higher-level mathematical thinking. They involve seeing connections and predicting what comes next, based on a sequence or rule.

Patterns: These may be numerical or geometric, allowing us to anticipate the next element in a sequence.

1, 2, 4, 8 … (Doubling each time)

Relationships: Understanding that certain pieces of information are related, and using this to solve problems, is key to mathematical reasoning . An example might be in understanding that if a car covers more distance in an hour at a faster speed, there’s a direct relationship between speed and distance.

In fostering these fundamental skills, we offer our learners not just theory but engaging challenges through mathematical puzzles and games that reinforce these concepts. By participating in such activities, children can apply these foundational maths skills, making connections between theory and practice in a dynamic and enjoyable way.

Teaching Mathematics Effectively

In our quest to develop sharp problem solvers and creative thinkers, we must focus on how mathematics is taught in primary schools. Our approach bridges the gap between traditional teaching and innovative, engaging methods .

Role of Teachers

We, as educators, play a pivotal role in moulding students into proficient problem-solvers. By incorporating a variety of maths puzzles and games into our classrooms, we make abstract concepts tangible and accessible. We believe in adapting our teaching styles to meet individual learning needs, devising strategies that make mathematics more than just a subject – instead, a captivating journey of discovery.

Curriculum Integration

The curriculum should not solely dictate our teaching, but serve as a dynamic framework that integrates real-world problem-solving and logical reasoning through mathematical activities . It is essential we weave challenging exercises with relevant contexts into lessons, thus aligning with our commitment to a broad and balanced education where every child flourishes.

The Importance of Problem-Solving Skills

Problem-solving skills are integral to learning mathematics, as they enable children to approach complex problems strategically and with confidence. Our goal is to harness these skills through puzzles and games that challenge and engage primary school children.

Developing Critical Thinking

We understand that critical thinking is the foundation of effective problem-solving. Our puzzles and games are designed to stimulate students’ thinking and reasoning processes , encouraging them to make connections and derive solutions based on logical deduction.

Cultivating Patience and Persistence

Throughout the problem-solving journey, we foster a sense of patience and persistence in our young learners. Recognising that solutions are not always immediate or straightforward, our resources teach children the value of perseverance and resilience in the face of challenging tasks.

Engaging Pupils with Interactive Maths Games

Interactive maths games are a brilliant way for us to make mathematics appealing and accessible to primary school pupils. By selecting the right games and integrating 21st-century technology, we can transform the learning experience into an interactive adventure that captivates our young learners.

Selecting Appropriate Games

When choosing maths games, we must ensure that they are appropriately challenging and aligned with the curriculum. The games should be designed to encourage pupils to think critically and develop their problem-solving skills. We often look for options that offer multiple levels of difficulty, which enables us to cater to the diverse abilities within a classroom. The aim is to select games that not only educate but also genuinely engage the students, making the learning process both enjoyable and effective.

Integrating ICT in Maths Games

In today’s digital age, integrating Information and Communication Technology (ICT) in maths games is not just innovative; it’s essential. We leverage ICT to provide simulations and virtual environments where pupils can explore mathematical concepts. This interactive technology helps to create immersive experiences that make abstract ideas more concrete. Using ICT, we can give pupils the chance to practise and hone their maths skills through fun, tech-driven games, fostering a more dynamic and interactive learning environment.

Incorporating interactive maths games within the classroom is a testament to our commitment to engaging and educating our pupils in a way that resonates with their experiences and interests. Through careful selection and the use of modern ICT, we are able to provide a learning experience that is both enriching and exciting.

Designing Challenging Maths Puzzles

In our pursuit to build effective problem-solvers, we focus on introducing maths puzzles that are engaging and non-routine, designed to foster natural curiosity in learners.

Creating Non-Routine Tasks

We believe that the essence of problem-solving lies in the ability to tackle non-routine tasks. These are not your everyday textbook problems, but instead, they present a scenario that requires students to apply concepts in ways they might not have anticipated. In constructing these tasks, it’s vital to strike a balance—too easy and they won’t push the envelope; too hard and they might discourage learners. For example, a puzzle involving pattern recognition might ask pupils to identify the underlying rule of a sequence and predict the next set of numbers.

Fostering Curiosity Through Puzzles

Curiosity drives us to explore the unknown, and with maths puzzles, it translates into learners venturing beyond their comfort zones. We craft puzzles that inherently provoke students’ interest and pique their natural inquisitiveness. A well-thought-out puzzle can act as an open inquiry, where the journey to the solution is as valuable as the solution itself. Consider a jigsaw arithmetic puzzle that requires learners to not only solve for missing pieces but also to understand why those pieces fit together as they do.

Incorporating Diverse Mathematical Tasks

When we introduce a variety of mathematical tasks to primary students, we lay the foundation for robust problem-solving skills. By engaging in an assortment of challenges , children can develop a deeper understanding of numbers, spatial awareness, and logical reasoning.

Exploration and Investigation

We believe that learning should be an adventure, where exploration and investigation play pivotal roles. By offering tasks that encourage students to explore, we open up opportunities for hands-on learning and inquisitive thinking. For instance, we might ask them to investigate the number of ways to reach a total of 10 using only red and blue counters. This simple task invites them to explore addition and the concept of combinations.

Measurement and Geometry

Measurement and geometry are two areas where children can apply maths to real-world scenarios. We often ask our students to measure lengths and widths of classroom objects, using rulers and other tools to relate the numbers to physical attributes. Then, we might move on to explore geometric shapes by examining and constructing models , allowing for a tangible understanding of edges, faces, and vertices. This hands-on experience is invaluable for cementing their conceptual knowledge.

Overcoming Obstacles in Problem Solving

Engaging with maths puzzles and games in primary education can sometimes present challenges. It’s essential we understand these stumbling blocks and employ effective strategies to help our children become confident problem solvers.

Identifying Common Mistakes

Often, obstacles in problem solving arise from common misunderstandings or repeated mistakes . We might see children rush through a problem without fully understanding it, or they may become fixed on one approach and not consider alternative methods. Documenting these mistakes offers us a chance to address them directly.

• Rushing : Not taking enough time to understand the problem.
• Fixation on one strategy : Failing to consider different angles.
• Overlooking details : Missing out on crucial information within the problem.

Strategies to Overcome Challenges

After pinpointing the typical mistakes, our strategies should focus on overcoming these barriers and reinforcing effective problem solving habits.

• Encourage thorough reading : Urge pupils to read problems several times.
• Promote multiple approaches : Introduce a variety of methods to tackle a single problem.
• Detail orientation : Teach children to pay attention to all the information given.

Remember, practice makes perfect, and providing children with a mixture of puzzles and games can build their resilience and adaptability in maths.

Creating a Supportive Maths Culture

At the heart of nurturing future problem-solvers is the establishment of a supportive maths culture . This means creating an environment where every child feels valued and capable of mastering mathematical challenges .

Encouraging Open Communication

We recognise the importance of open communication in the classroom. It’s crucial to foster an atmosphere where pupils feel comfortable to express their ideas, ask questions, and share their experiences. In our classroom, we encourage learners to articulate their thought processes and reasoning. This not only clarifies their understanding but also enriches peer learning, as students learn from each other’s perspectives. Such a culture not only enhances their communication skills but also demystifies complex concepts, making maths more approachable .

Building Confidence and Growth Mindset

We strive to infuse our pupils with confidence and a growth mindset . To do this, we emphasise the belief that abilities can be developed through dedication and hard work. This contrasts with a fixed mindset, where children might believe their skills are static and unchangeable. By celebrating effort rather than innate ability , we inspire our students to embrace challenges and learn from mistakes—an approach that is pivotal for fostering resilient problem solvers . Our aim is to show that mathematics is not about being ‘right’ all the time; instead, it’s about the adventure of learning and improving.

Assessing and Enhancing Problem-Solving Capabilities

In our quest to build adept problem-solvers through maths puzzles and games , it’s vital that we effectively assess and enhance primary students’ problem-solving capabilities. This not only involves gauging their current skill levels but also providing constructive feedback to aid their growth.

Conducting Meaningful Assessments

When it comes to assessing problem-solving activities, our approach is to create evaluations that are as engaging as the learning experiences themselves. We believe that a student’s problem-solving skills are best understood by observing them during actual problem-solving activities. Such assessments might involve practical tasks where pupils apply heuristics or use visual representations to tackle mathematical problems, as highlighted by Developing Mathematical Problem-Solving Skills . These tasks are designed to mirror ‘real life’ scenarios, requiring students to think critically and creatively.

Providing Constructive Feedback

Feedback is a powerful tool in our educational arsenal—it informs students about their performance and provides guidance on how they can improve. Our feedback is specific, timely, and always focused on strategies that students can use to enhance their problem-solving abilities. Whether it’s encouraging greater representational fluency or exploring diverse strategies, we ensure that our feedback helps students reflect and grow as problem solvers. By drawing on resources such as the Real Engagement in Active Problem Solving (REAPS) model, we support teachers in offering feedback that fosters creative problem solving in mathematics.

In all of our endeavours at LearningMole, we strive to provide learning experiences that not only educate but also excite. We’re dedicated to fostering environments where every student can become a confident and proficient problem solver.

Resources for Primary Maths Education

We know that the right resources can make all the difference in empowering primary maths education. So, let’s explore some excellent materials that you can access for free, as well as delve into the treasures offered by NRICH to enrich our young learners’ mathematical journey.

Free Teaching Materials

We pride ourselves on providing a variety of free teaching materials designed to make maths engaging and fun . You’ll find interactive tutorials , activity sheets , and articles, all tailored for the curious minds of primary children. These resources are not only educational, but they also allow children to absorb key maths concepts in a way that feels like play.

• Interactive Tutorials : Engage with our step-by-step guides that bring clarity to complex maths problems.
• Articles : Discover insights and tips targeted at enhancing the teaching experience.
• Activity Sheets : Download and print these for hands-on practice that reinforces mathematical understanding.

One platform we admire is LearningMole , which offers a wealth of content to help children discover the joy of maths through various fun and creative resources.

Utilising NRICH Resources

When it comes to using NRICH resources , we’re looking at a treasure trove of maths puzzles and activities that are perfect for primary pupils. NRICH aims to challenge and excite young minds with games that are both intriguing and highly educational.

• Challenging Puzzles : They stimulate strategic thinking and offer varying degrees of difficulty to suit all levels.
• Classroom Activities : Carefully crafted to promote collaborative problem-solving among students.

These resources from NRICH not only complement our teaching but also bring a new dimension to primary maths, making our lessons more dynamic and effective.

Frequently Asked Questions

In this section, we’ll address some of the most common inquiries related to enhancing problem-solving skills through mathematics puzzles and games in primary education.

What are some engaging activities that can enhance problem-solving abilities in primary school children?

Children in primary school can greatly benefit from activities such as building structures using different shapes or collaborative tasks where they devise solutions to “real life” scenarios. These activities encourage them to apply mathematical concepts in practical ways.

Can you suggest some games that improve problem-solving skills for young learners?

Certainly! Games like chess, Sudoku, and even certain board games that require strategic thinking can improve problem-solving skills . These games challenge young minds to think ahead and plan their moves carefully.

How can puzzles be effectively used to develop mathematical problem-solving competencies in children?

Puzzles can be used in teaching by posing them as interesting challenges that are mathematically meaningful . Puzzles such as tangrams, magic squares, and logic puzzles encourage children to use mathematical reasoning and pattern recognition.

In what ways can teachers incorporate problem-solving exercises into their primary classroom curriculum?

Teachers can introduce problem-solving in the classroom by integrating puzzles and games into lesson plans. They might also ask thought-provoking questions that lead to problem-solving discussions or use differentiated instruction to cater to various learning styles.

Which hands-on group activities can help children build teamwork and problem-solving skills concurrently?

Group activities that involve building projects from common objects like blocks or recycled materials allow children to work together. They can also participate in team-based challenges that require collective problem-solving and decision-making.

What types of maths challenges are suitable for primary students to promote critical thinking and reasoning?

Suitable challenges include pattern identification , sequencing tasks, basic arithmetic puzzles, and solving mathematical puzzles . These can be tailored to match the students’ age and proficiency level, ensuring the tasks remain engaging and appropriately challenging.

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3 Math Tasks You Already Use That Can Foster Collaboration

Elementary teachers can turn common activities into opportunities for students to work and learn together.

Picture a “traditional” math class and then a “21st-century” math class. One of the most obvious differences is probably how students are interacting: No longer silently sitting in rows of desks, students are talking to each other. They are collaborating. 

Collaborative learning promotes higher-level reasoning, self-esteem, and positive relationships at school . To unlock the benefits of collaborative learning for young mathematicians, teachers can plan accessible tasks. When students don’t have to devote as much “brain space” to understanding the directions for a task, they can focus instead on collaborating on the math itself. This supports all students, including multilingual students and students with disabilities.

Start with tasks that are engaging, but not content-heavy , so that you can prioritize laying the groundwork for collaboration. Even a task as simple as taking turns drawing a picture can be productive for discussing expectations around sharing work and communicating. From there, you can use the warm-ups, word problems, and manipulatives that you and your students are already familiar with to generate accessible and meaningful collaborative tasks.

1. Collaborate Using Your Warm-Ups

Routines like warm-ups, number sense routines, or number talks invite students to reason and discuss, typically in a whole group setting. However, these can be done collaboratively as well. 

Selecting warm-ups for collaboration: Look for routines that have many possible “right” answers, so that all group members will be able to contribute meaningfully. This doesn’t have to be multiple numerical answers; it could also be many ways to get to a solution—for example, different ways to see a total in a dot talk . When you do this routine as a whole group, you may find yourself saying to students, “Wow, I hadn’t thought of that!” Those activities lend themselves well to collaboration.

Preparing warm-ups for collaboration: In a whole group discussion, you use your learning objectives to steer the conversation as it happens. For example, if we were discussing “Ways to Make 1/8” as a whole class and students were only talking about shapes, I might ask the class to think about a number line. If this discussion were to happen in collaborative groups, that steering needs to be built into the task. One way to do this is with a checklist. For example, when my third graders did “Ways to Make 1/8” as a collaborative task, each group got this checklist:

• 2 ways to make 1/8 of a square
• 2 ways to make 1/8 of a rectangle
• 1 way to make 1/8 of a circle
• 1 way to make 1/8 on a number line
• 2 ways to make 1/8 of numbers (example: 1/8 of 16 is 2)

The checklist makes it obvious to students how they should be contributing, and even though I might not be part of every conversation, I know that key points are surfacing throughout the room.

2. Collaborate Using Your Word Problems

Whether writing word problems yourself or using the ones in a curriculum, word problems are a feature of every math class.

Selecting word problems for collaboration: Make sure the context of the chosen word problem is accessible, or plan for making the context accessible. You might use images or videos to make sure students understand the vocabulary and can make connections to background knowledge. If the word problems in your curriculum have instructions like “Use an array model to solve,” leave those parts out. Groups should solve in ways that make sense to them, and diversity in strategies both within groups and between groups will lead to important mathematical discussions.

Preparing word problems for collaboration: You can turn one word problem into an extended collaborative task by planning a series of increasingly challenging number sets to fit into the story. For example, “Allison filled up 8 baskets of peaches, with 10 peaches in each basket. How many peaches is that?” followed by 18 baskets of 10, then 18 baskets of 25. If groups are working at vertical surfaces like whiteboards, they can look around the room after finishing one number set to see what number set to work on next, as described by Peter Liljedahl in Building Thinking Classrooms . This promotes knowledge mobility and frees the teacher to discuss with groups as needed. 

3. Collaborate Using Your Manipulatives

Counting collections is a rich collaborative task for students across elementary grades. You might be surprised by how excited students are to count even the most familiar math manipulatives. 

Preparing counting collections: Students will need “right-sized” collections, tools for organizing, and space to record. Don’t be afraid to offer big collections. Collections of, say, 45 for kindergartners or 845 for fourth graders can be meaningful opportunities for developing place value understanding. Tools like cups, plates, paper trays from the cafeteria, and your number lines or hundred charts should be available for students to take up as they need. Finally, think about how you want students to record their collections: on paper , on whiteboards, and/or digitally on a platform like Seesaw.

To grow your collection of collections beyond your classroom manipulatives and supplies, invite students to bring in collections from home, or round up unwanted items around your school. I have many beautiful collections of transparent manipulatives left over from the overhead projector days.

If At First You Don’t Succeed…

Collaboration is tricky. Even with extensive planning and coaching , students and groups will struggle at times. But if collaboration is hard for your class , that means they need to practice it more, not less. More collaboration doesn’t have to mean planning more tasks from scratch; you can and should use the resources you already have and turn those into revisit-able routines. Having familiar routines for collaboration means that students can dive straight into the learning, while continuing to develop their identities as good group-mates who know how to support each other and work together. 

Problem Solving in Mathematics Education

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• 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
• Prospective Teacher
• 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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Singer, F. M., Ellerton, N. F., & Cai, J. (Eds.). (2015). Mathematical problem posing. From research to practice . NY: Springer.

Törner, G., Schoenfeld, A. H., & Reiss, K. M. (2007). Problem solving around the world: Summing up the state of the art. ZDM—The International Journal on Mathematics Education, 39 (1), 5–6.

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Watson, A., & Ohtani, M. (2015). Themes and issues in mathematics education concerning task design: Editorial introduction. In A. Watson & M. Ohtani (Eds.), Task design in mathematics education, an ICMI Study 22 (pp. 3–15). NY: Springer.

Zimmermann, B. (1983). Problemlösen als eine Leitidee für den Mathematikunterricht. Ein Bericht über neuere amerikanische Beiträge. Der Mathematikunterricht, 3 (1), 5–45.

Further Reading

Boaler, J. (1997). Experiencing school mathematics: Teaching styles, sex, and setting . Buckingham, PA: Open University Press.

Borwein, P., Liljedahl, P., & Zhai, H. (2014). Mathematicians on creativity. Mathematical Association of America.

Burton, L. (1984). Thinking things through . London, UK: Simon & Schuster Education.

Feynman, R. (1999). The pleasure of finding things out . Cambridge, MA: Perseus Publishing.

Gardner, M. (1978). Aha! insight . New York, NY: W. H. Freeman and Company.

Gardner, M. (1982). Aha! gotcha: Paradoxes to puzzle and delight . New York, NY: W. H. Freeman and Company.

Gardner, H. (1993). Creating minds: An anatomy of creativity seen through the lives of Freud, Einstein, Picasso, Stravinsky, Eliot, Graham, and Ghandi . New York, NY: Basic Books.

Glas, E. (2002). Klein’s model of mathematical creativity. Science & Education, 11 (1), 95–104.

Hersh, D. (1997). What is mathematics, really? . New York, NY: Oxford University Press.

Root-Bernstein, R., & Root-Bernstein, M. (1999). Sparks of genius: The thirteen thinking tools of the world’s most creative people . Boston, MA: Houghton Mifflin Company.

Zeitz, P. (2006). The art and craft of problem solving . New York, NY: Willey.

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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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Title: mathify: evaluating large language models on mathematical problem solving tasks.

Abstract: The rapid progress in the field of natural language processing (NLP) systems and the expansion of large language models (LLMs) have opened up numerous opportunities in the field of education and instructional methods. These advancements offer the potential for tailored learning experiences and immediate feedback, all delivered through accessible and cost-effective services. One notable application area for this technological advancement is in the realm of solving mathematical problems. Mathematical problem-solving not only requires the ability to decipher complex problem statements but also the skill to perform precise arithmetic calculations at each step of the problem-solving process. However, the evaluation of the arithmetic capabilities of large language models remains an area that has received relatively little attention. In response, we introduce an extensive mathematics dataset called "MathQuest" sourced from the 11th and 12th standard Mathematics NCERT textbooks. This dataset encompasses mathematical challenges of varying complexity and covers a wide range of mathematical concepts. Utilizing this dataset, we conduct fine-tuning experiments with three prominent LLMs: LLaMA-2, WizardMath, and MAmmoTH. These fine-tuned models serve as benchmarks for evaluating their performance on our dataset. Our experiments reveal that among the three models, MAmmoTH-13B emerges as the most proficient, achieving the highest level of competence in solving the presented mathematical problems. Consequently, MAmmoTH-13B establishes itself as a robust and dependable benchmark for addressing NCERT mathematics problems.

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OpenAI’s ‘Strawberry’ AI Set to Launch in October with Complex Math and Programming Abilities

OpenAI's new AI model, Strawberry, will focus on complex tasks like math and programming, addressing limitations of current systems.

OpenAI is planning to roll out a new artificial intelligence model, named ‘Strawberry,' aimed at enhancing chatbots ' reasoning abilities. According to a report from The Information , the model will be released during October. The AI focuses on tackling intricate tasks such as advanced mathematics and programming , areas where current systems have limitations.

Rebranding and Technical Capabilities

Originally named Q* , the model has been renamed ‘Strawberry.' The new AI is set to enhance OpenAI's present systems considerably. Sources told The Information that OpenAI presented ‘Strawberry' to U.S. national security officials during the summer, pointing to its potential applications in that field.

Earlier this month, OpenAI CEO Sam Altman hinted at the new AI model by sharing an image of strawberries. The cryptic message became clear with the official unveiling of ‘Strawberry.' OpenAI's model is expected to bolster chatbot capabilities, allowing them to undertake tasks that require a higher level of reasoning and problem-solving.

Strategic Importance and Future Prospects

Introducing ‘Strawberry' is part of OpenAI's strategy to advance conversational AI. By addressing the shortcomings of current models, ‘Strawberry' aims to improve chatbot reliability and functionality across various applications. The move underscores OpenAI's commitment to remaining competitive in the ever-evolving tech sector.

OpenAI is also working on another AI model named ‘Orion,' which aims to exceed GPT-4's capabilities, with ‘Strawberry' playing a significant role. Insiders indicate that a chatbot version of ‘Strawberry' might debut this fall, potentially integrated into ChatGPT . The model is designed to solve complex math problems and optimize programming tasks, with enhanced logic that improves language-related challenge-solving.

Internal Testing and Performance

In internal trials, ‘Strawberry' demonstrated its prowess by solving the New York Times word puzzle “Connections.” An AI tested internally by OpenAI, likely ‘Strawberry,' scored over 90 percent on the MATH benchmark . Internal documents reveal plans to use ‘Strawberry' for autonomous internet searches, enabling it to conduct comprehensive research and planning.

The development approach for ‘Strawberry' is similar to Stanford's “Self-Taught Reasoner” (STaR) method , designed to boost AI reasoning skills. Former OpenAI chief researcher Ilya Sutskever, now running a startup focused on secure super AI, is credited with providing the foundational ideas for ‘Strawberry.'

Google DeepMind is also delving into AI systems with advanced mathematical abilities, having created AlphaProof and AlphaGeometry 2 , which achieved silver at the International Mathematical Olympiad. However, it remains unclear how scalable and generalizable these models will be.

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What is JUMP Math, and why are some teachers raving about it? Try 13 of its brain-teasing problems to find out

Michelle Jones says her students have been far more engaged since she started using the JUMP Math method at Jarvis Traditional Elementary School in Delta, B.C. Try the quiz at the end of this article to see whether you could solve the problems that Ms. Jones's Grade 6 students are expected to handle. Jennifer Gauthier/The Globe and Mail

In her more than two decades in front of a classroom, Michelle Jones has used five different math textbooks and, until recently, had grown increasingly frustrated in her inability to reach many of her students.

The story-based math problems that filled those textbooks left most of the children either checked out or confused. She’d draw on videos and other materials to supplement her lessons, but it didn’t feel like that was enough to help her students build their confidence in the subject.

Then, three years ago, her board – the Delta School District in Delta, B.C. – piloted a program that incorporated JUMP Math, a resource originally developed by John Mighton, an accomplished playwright and entrepreneur in Toronto. The program, run by a charity established in 2002, emphasizes students rehearsing basic arithmetic operations so they can see patterns and break problems down into smaller parts, gradually raising the level of difficulty. “We were just ready for a shift, to try something different,” said Ms. Jones, who teaches Grades 6 and 7 at Delta’s Jarvis Traditional Elementary School.

John Mighton helps Grade 4 pupils practice JUMP techniques at a Toronto school in 2007, five years after he developed the method. Deborah Baic/The Globe and Mail

For Ms. Jones, using JUMP Math – JUMP stands for Junior Undiscovered Math Prodigies – represented a sea change.

She is now armed with new strategies to teach the subject and has been able to reintroduce some rote learning so that students can engage with the material more quickly. The students use whiteboards, check in with their partners and practise on their own. As a result, she’s noticed they are less anxious and take more risks in class.

“In my experience, I have never seen students so engaged, relaxed and enjoying a math lesson,” she said.

The pilot at Delta has since expanded: In 14 of the district’s 24 elementary schools, most of the teachers are now using JUMP.

Focusing on math fact fluency may seem like an obvious recipe for success, but the way math is taught in schools has been the subject of a long-standing and divisive debate, much like reading.

On one side, some experts and educators believe rote learning creates anxiety and dread, and that children should approach the subject with playfulness and curiosity by learning through problem solving, pattern discovery and open-ended exploration.

Others have advocated for a so-called back-to-basics approach and pushed governments to initiate curriculum changes so students have the ability to quickly recall addition, subtraction, multiplication and division through repetition and memorization. Rote learning shouldn’t be considered a dirty phrase, they argue.

The debate comes at a critical time: Although Canada performs well compared with other countries globally, Canadian students’ scores on an international test administered by the Organization for Economic Co-operation and Development have been slipping for almost two decades – and the latest results from late last year show that slide continuing.

Neil Stephenson, director of learning services at Delta, brought in JUMP Math because he felt something needed to change in his district.

Educators were doing a “hodgepodge of things” to help students meet curriculum expectations, he said, which put an incredible strain on them to find and build lesson plans.

After doing some research and finding JUMP, he approached an elementary school that hadn’t been scoring well on provincial tests to see if any teachers there would try the program. Around three-quarters of them raised their hands – and assessments at the end of that school year showed that several students had progressed multiple grade-levels, and teacher confidence in how they approach the subject rose, he said.

“Absolutely we want kids to be doing creative work and solving interesting questions and synthesizing their knowledge. But there has to be some building up of that knowledge somewhere else first,” Mr. Stephenson said.

In Jarvis Elementary's district, more than half the schools now use JUMP for math education. Jennifer Gauthier/The Globe and Mail

That is heartening to JUMP’s founder.

Mr. Mighton didn’t fare well in math in school and nearly failed first-year calculus in university. But he slowly overcame his own math anxiety and, as a playwright trying to make a living, started tutoring the subject later in life. Teaching children encouraged him to break down difficult concepts into smaller parts, and, in turn, grasp the subject better. He relearned concepts he had missed along the way, and then returned to school in his early 30s to earn a PhD in math at the University of Toronto.

“Math is actually accessible, very accessible,” he said.

He explained that the current method – investigating ideas through problem solving, pattern discovery and open-ended exploration – rushes children past learning math facts in the hopes of making the subject more engaging. It has the opposite effect, he said, because children actually just become confused and disengaged.

His program provides lesson plans for teachers that allows for an incremental approach to problem solving. There’s a workbook for students, but Mr. Mighton said that should only be used after the lessons. “You want to get to those problems, but that’s not where you start. That’s the mistake we’re making,” he said. “We always think kids are experts. And we give them problems that are designed for experts when they’re novice learners.”

Math professor Anna Stokke feels that methods of teaching introduced in the 1980s have done a 'disservice to children' in the decades since. John Woods/the Globe and Mail

Anna Stokke, a mathematics professor at the University of Winnipeg and a vocal proponent for schools to once again focus on fundamentals, said the change in how math was taught began in the late 1980s under the school of thought called constructivism. The theory suggests students should not passively acquire knowledge through direct instruction but rather learn through experiences and interactions. At the time, the National Council of Teachers of Mathematics in the U.S. released a set of standards where problem solving became the focus of instruction, she said. The movement then spread to Canada.

Prof. Stokke said the change in instruction has been a “disservice to children” because students should be practising math procedures and memorizing facts before they can grasp more complex problems. “I’m a mathematician and, believe me, I know how to solve complex problems. And you can’t do complex problems without having a web of knowledge in your brain.”

The result of this change has been a widening equity gap, she said, where families who have the means provide tutoring for their children, while others continue to struggle in the subject.

However, Jason To, a math co-ordinator at the Toronto District School Board, said the argument that schools are teaching one way over another is misplaced. He worries that some experts are latching onto international test scores and insinuating that inquiry-based instruction is dominating the education space. But teachers, he said, are doing both: instructing their students on math fluency and immersing them in complex problems.

“This debate to me is you got to do one versus the other, and it’s not productive. It’s more like, how do these co-exist?”

Math fluency, and the way it is measured in standardized test, can be polarizing subjects in the world of education. Justin Tang/The Globe and Mail

Janelle Feenan, a teacher and peer support co-ordinator at the Delta school division, echoed the sentiment. For years, she and her Grade 3 teacher colleague would spend an evening a week researching and pulling resources to help their students with math fluency and to develop a more comprehensive understanding for concepts.

“We were struggling a little bit to make sure our students were understanding what we were doing. We’re going through the motions, but we just didn’t feel that they were where they needed to be,” she said.

They raised their hands to participate in the pilot that introduced JUMP Math to students.

Having worked with the program, Ms. Feenan found that there’s a place for both the structural approach that JUMP provides as well as allowing for problem solving and conceptual understanding. She uses JUMP as her main lesson plan, and then supports that with games and visual aids to deepen understanding.

“Neither of those approaches alone would be adequate to prepare kids for success in math,” she said. “I think you have to supplement lessons with activities and resources that are fun and engaging to build their understanding and enrich their learning experience.”

Pop quiz: Test your math skills, JUMP-style

These are Grade 6-level problems from JUMP Math assessment and practice books. Get out your calculator app and give them a try!

c. If the pitcher pitches on the first game (or on the second, or on the third), she will pitch a total of 10 games, ending on the 46th game (or 47th, or 48th, respectively).

Photo: Jon Blacker/The Canadian Press

d. The lake with the longest shoreline is Huron, at 6,164 km. The shortest is Lake Ontario, 1,146 km. The difference is 6,164 – 1,146 = 5,018 km.

a. Avril’s grade sold 10 + 15 + 25 + 10 = 60 tickets in total. Of those 60 tickets, 30 (half) are adult tickets and sell for $5 each, and the other 30 sell for $3 each. So Avril’s grade raises (30 × $5) + (30 × $3) = $150 + $90 = $240. Since the bus costs $320, there is still $320 – $240 = $80 needed.

c. Round 3,128 to 3,000, and 4,956 to 5,000. So 3,128 × 4,956 is approximately equal to 3,000 × 5,000 = 15,000,000, i.e., 15 million.

b. 821 × 4 = 3,284. To calculate mentally, multiply the digits separately.

c. The perimeter of the field is 921 × 5 = 4,605 m. The farmer needs 4,605 – 4,500 = 105 more metres of fence to surround the field.

Photo illustration (source: Ina Fassbender/AFP/Getty Images, JUMP Math

b. Add the digits and check if the sum makes a multiple of nine.

c. 40 per cent of 20 = 8, and 25 per cent of 20 = 5. Since 8 + 5 = 13, there are 20 – 13 = 7 green fish.

a. $7.21 × 3 = $21.63. To multiply mentally, multiply the digits separately.

Photo: Kham/Reuters

d. 84.8 mm ÷ 4 = 21.2 mm. To divide mentally, divide the digits separately.

b. 220 kg ÷ 4 = 55 kg.

c. 15 × 8 = 120, 120 ÷ 100 = 1.20. They will pay $1.20 in taxes.

d. Three quarters of 12 is nine, so nine green balloons have writing on them. Sixty per cent of 15 is nine, so nine blue balloons have writing on them. So, 9 + 9 = 18 balloons in total have writing on them.

Photo: Vadim Ghirda/AP

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