what are the disadvantages of problem solving in teaching

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5 Advantages and Disadvantages of Problem-Based Learning [+ Activity Design Steps]

Written by Marcus Guido

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• Teaching Strategies

Advantages of Problem-Based Learning

Disadvantages of problem-based learning, steps to designing problem-based learning activities.

Used since the 1960s, many teachers express concerns about the effectiveness of problem-based learning (PBL) in certain classroom settings.

Whether you introduce the student-centred pedagogy as a one-time activity or mainstay exercise, grouping students together to solve open-ended problems can present pros and cons.

Below are five advantages and disadvantages of problem-based learning to help you determine if it can work in your classroom.

If you decide to introduce an activity, there are also design creation steps and a downloadable guide to keep at your desk for easy reference.

1. Development of Long-Term Knowledge Retention

Students who participate in problem-based learning activities can improve their abilities to retain and recall information, according to a literature review of studies about the pedagogy .

The literature review states “elaboration of knowledge at the time of learning” -- by sharing facts and ideas through discussion and answering questions -- “enhances subsequent retrieval.” This form of elaborating reinforces understanding of subject matter , making it easier to remember.

Small-group discussion can be especially beneficial -- ideally, each student will get chances to participate.

But regardless of group size, problem-based learning promotes long-term knowledge retention by encouraging students to discuss -- and answer questions about -- new concepts as they’re learning them.

2. Use of Diverse Instruction Types

You can use problem-based learning activities to the meet the diverse learning needs and styles of your students, effectively engaging a diverse classroom in the process. In general, grouping students together for problem-based learning will allow them to:

• Address real-life issues that require real-life solutions, appealing to students who struggle to grasp abstract concepts
• Participate in small-group and large-group learning, helping students who don’t excel during solo work grasp new material
• Talk about their ideas and challenge each other in a constructive manner, giving participatory learners an avenue to excel
• Tackle a problem using a range of content you provide -- such as videos, audio recordings, news articles and other applicable material -- allowing the lesson to appeal to distinct learning styles

Since running a problem-based learning scenario will give you a way to use these differentiated instruction approaches , it can be especially worthwhile if your students don’t have similar learning preferences.

3. Continuous Engagement

Providing a problem-based learning challenge can engage students by acting as a break from normal lessons and common exercises.

It’s not hard to see the potential for engagement, as kids collaborate to solve real-world problems that directly affect or heavily interest them.

Although conducted with post-secondary students, a study published by the Association for the Study of Medical Education reported increased student attendance to -- and better attitudes towards -- courses that feature problem-based learning.

These activities may lose some inherent engagement if you repeat them too often, but can certainly inject excitement into class.

4. Development of Transferable Skills

Problem-based learning can help students develop skills they can transfer to real-world scenarios, according to a 2015 book that outlines theories and characteristics of the pedagogy .

The tangible contexts and consequences presented in a problem-based learning activity “allow learning to become more profound and durable.” As you present lessons through these real-life scenarios, students should be able to apply learnings if they eventually face similar issues.

For example, if they work together to address a dispute within the school, they may develop lifelong skills related to negotiation and communicating their thoughts with others.

As long as the problem’s context applies to out-of-class scenarios, students should be able to build skills they can use again.

5. Improvement of Teamwork and Interpersonal Skills

Successful completion of a problem-based learning challenge hinges on interaction and communication, meaning students should also build transferable skills based on teamwork and collaboration . Instead of memorizing facts, they get chances to present their ideas to a group, defending and revising them when needed.

What’s more, this should help them understand a group dynamic. Depending on a given student, this can involve developing listening skills and a sense of responsibility when completing one’s tasks. Such skills and knowledge should serve your students well when they enter higher education levels and, eventually, the working world.

1. Potentially Poorer Performance on Tests

Devoting too much time to problem-based learning can cause issues when students take standardized tests, as they may not have the breadth of knowledge needed to achieve high scores. Whereas problem-based learners develop skills related to collaboration and justifying their reasoning, many tests reward fact-based learning with multiple choice and short answer questions. Despite offering many advantages, you could spot this problem develop if you run problem-based learning activities too regularly.

2. Student Unpreparedness

Problem-based learning exercises can engage many of your kids, but others may feel disengaged as a result of not being ready to handle this type of exercise for a number of reasons. On a class-by-class and activity-by-activity basis, participation may be hindered due to:

• Immaturity  -- Some students may not display enough maturity to effectively work in a group, not fulfilling expectations and distracting other students.
• Unfamiliarity  -- Some kids may struggle to grasp the concept of an open problem, since they can’t rely on you for answers.
• Lack of Prerequisite Knowledge  -- Although the activity should address a relevant and tangible problem, students may require new or abstract information to create an effective solution.

You can partially mitigate these issues by actively monitoring the classroom and distributing helpful resources, such as guiding questions and articles to read. This should keep students focused and help them overcome knowledge gaps. But if you foresee facing these challenges too frequently, you may decide to avoid or seldom introduce problem-based learning exercises.

3. Teacher Unpreparedness

If supervising a problem-based learning activity is a new experience, you may have to prepare to adjust some teaching habits . For example, overtly correcting students who make flawed assumptions or statements can prevent them from thinking through difficult concepts and questions. Similarly, you shouldn’t teach to promote the fast recall of facts. Instead, you should concentrate on:

• Giving hints to help fix improper reasoning
• Questioning student logic and ideas in a constructive manner
• Distributing content for research and to reinforce new concepts
• Asking targeted questions to a group or the class, focusing their attention on a specific aspect of the problem

Depending on your teaching style, it may take time to prepare yourself to successfully run a problem-based learning lesson.

4. Time-Consuming Assessment

If you choose to give marks, assessing a student’s performance throughout a problem-based learning exercise demands constant monitoring and note-taking. You must take factors into account such as:

• Completed tasks
• The quality of those tasks
• The group’s overall work and solution
• Communication among team members
• Anything you outlined on the activity’s rubric

Monitoring these criteria is required for each student, making it time-consuming to give and justify a mark for everyone.

5. Varying Degrees of Relevancy and Applicability

It can be difficult to identify a tangible problem that students can solve with content they’re studying and skills they’re mastering. This introduces two clear issues. First, if it is easy for students to divert from the challenge’s objectives, they may miss pertinent information. Second, you could veer off the problem’s focus and purpose as students run into unanticipated obstacles. Overcoming obstacles has benefits, but may compromise the planning you did. It can also make it hard to get back on track once the activity is complete. Because of the difficulty associated with keeping activities relevant and applicable, you may see problem-based learning as too taxing.

If the advantages outweigh the disadvantages -- or you just want to give problem-based learning a shot -- follow these steps:

1. Identify an Applicable Real-Life Problem

Find a tangible problem that’s relevant to your students, allowing them to easily contextualize it and hopefully apply it to future challenges. To identify an appropriate real-world problem, look at issues related to your:

• Students’ shared interests

You must also ensure that students understand the problem and the information around it. So, not all problems are appropriate for all grade levels.

2. Determine the Overarching Purpose of the Activity

Depending on the problem you choose, determine what you want to accomplish by running the challenge. For example, you may intend to help your students improve skills related to:

• Collaboration
• Problem-solving
• Curriculum-aligned topics
• Processing diverse content

A more precise example, you may prioritize collaboration skills by assigning specific tasks to pairs of students within each team. In doing so, students will continuously develop communication and collaboration abilities by working as a couple and part of a small group. By defining a clear purpose, you’ll also have an easier time following the next step.

3. Create and Distribute Helpful Material

Handouts and other content not only act as a set of resources, but help students stay focused on the activity and its purpose. For example, if you want them to improve a certain math skill , you should make material that highlights the mathematical aspects of the problem. You may decide to provide items such as:

• Data that helps quantify and add context to the problem
• Videos, presentations and other audio-visual material
• A list of preliminary questions to investigate

Providing a range of resources can be especially important for elementary students and struggling students in higher grades, who may not have self-direction skills to work without them.

4. Set Goals and Expectations for Your Students

Along with the aforementioned materials, give students a guide or rubric that details goals and expectations. It will allow you to further highlight the purpose of the problem-based learning exercise, as you can explain what you’re looking for in terms of collaboration, the final product and anything else. It should also help students stay on track by acting as a reference throughout the activity.

5. Participate

Although explicitly correcting students may be discouraged, you can still help them and ask questions to dig into their thought processes. When you see an opportunity, consider if it’s worthwhile to:

• Fill gaps in knowledge
• Provide hints, not answers
• Question a student’s conclusion or logic regarding a certain point, helping them think through tough spots

By participating in these ways, you can provide insight when students need it most, encouraging them to effectively analyze the problem.

6. Have Students Present Ideas and Findings

If you divided them into small groups, requiring students to present their thoughts and results in front the class adds a large-group learning component to the lesson. Encourage other students to ask questions, allowing the presenting group to elaborate and provide evidence for their thoughts. This wraps up the activity and gives your class a final chance to find solutions to the problem.

Wrapping Up

The effectiveness of problem-based learning may differ between classrooms and individual students, depending on how significant specific advantages and disadvantages are to you. Evaluative research consistently shows value in giving students a question and letting them take control of their learning. But the extent of this value can depend on the difficulties you face.It may be wise to try a problem-based learning activity, and go forward based on results.

Create or log into your teacher account on Prodigy -- an adaptive math game that adjusts content to accommodate player trouble spots and learning speeds. Aligned to US and Canadian curricula, it’s used by more than 350,000 teachers and 10 million students. It may be wise to try a problem-based learning activity, and go forward based on results.

• Effective Teaching Strategies

Problem-Based Learning: Benefits and Risks

• November 12, 2009
• Maryellen Weimer, PhD

Problem-based learning, the instructional approach in which carefully constructed, open-ended problems are used by groups of students to work through content to a solution, has gained a foothold in many segments of higher education.

Originally PBL, as it’s usually called, was used in medical school and in some business curricula for majors. But now it is being used in a wide range of disciplines and with students at various educational levels. The article (reference below) from which material is about to be cited “makes a critical assessment” of how PBL is being used in the field of geography.

Much of the content is relevant to that discipline specifically, but the article does contain a useful table that summarizes the benefits and risks of PBL for students, instructors, and institutions. Material on the table is gleaned from an extensive review of the literature (all referenced in the article). Here’s some of the information contained in the table.

Benefits of Problem-Based Learning

For Students

• It’s a student-centered approach.
• Typically students find it more enjoyable and satisfying.
• It encourages greater understanding.
• Students with PBL experience rate their abilities higher.
• PBL develops lifelong learning skills.

For Instructors

• Class attendance increases.
• The method affords more intrinsic reward.
• It encourages students to spend more time studying.
• It promotes interdisciplinarity.

For Institutions

• It makes student learning a priority.
• It may aid student retention.
• It may be taken as evidence that an institution values teaching.

Risks of Problem-Based Learning

• Prior learning experiences do not prepare students well for PBL.
• PBL requires more time and takes away study time from other subjects.
• It creates some anxiety because learning is messier.
• Sometimes group dynamics issues compromise PBL effectiveness.
• Less content knowledge may be learned.
• Creating suitable problem scenarios is difficult.
• It requires more prep time.
• Students have queries about the process.
• Group dynamics issues may require faculty intervention.
• It raises new questions about what to assess and how.
• It requires a change in educational philosophy for faculty who mostly lecture.
• Faculty will need staff development and support.
• It generally takes more instructors.
• It works best with flexible classroom space.
• It engenders resistance from faculty who question its efficacy.

Reference: Pawson, E., Fournier, E., Haight, M., Muniz, O., Trafford, J., and Vajoczki, S. 2006. Problem-based learning in geography: Towards a critical assessment of its purposes, benefits and risks. Journal of Geography in Higher Education 30 (1): 103–16.

Excerpted from The Teaching Professor , February 2007.

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• Establishing Community Agreements and Classroom Norms
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Problem-Based Learning

Problem-based learning  (PBL) is a student-centered approach in which students learn about a subject by working in groups to solve an open-ended problem. This problem is what drives the motivation and the learning. 

Why Use Problem-Based Learning?

Nilson (2010) lists the following learning outcomes that are associated with PBL. A well-designed PBL project provides students with the opportunity to develop skills related to:

• Working in teams.
• Managing projects and holding leadership roles.
• Oral and written communication.
• Self-awareness and evaluation of group processes.
• Working independently.
• Critical thinking and analysis.
• Explaining concepts.
• Self-directed learning.
• Applying course content to real-world examples.
• Researching and information literacy.
• Problem solving across disciplines.

Considerations for Using Problem-Based Learning

Rather than teaching relevant material and subsequently having students apply the knowledge to solve problems, the problem is presented first. PBL assignments can be short, or they can be more involved and take a whole semester. PBL is often group-oriented, so it is beneficial to set aside classroom time to prepare students to   work in groups  and to allow them to engage in their PBL project.

Students generally must:

• Examine and define the problem.
• Explore what they already know about underlying issues related to it.
• Determine what they need to learn and where they can acquire the information and tools necessary to solve the problem.
• Evaluate possible ways to solve the problem.
• Solve the problem.
• Report on their findings.

Getting Started with Problem-Based Learning

• Articulate the learning outcomes of the project. What do you want students to know or be able to do as a result of participating in the assignment?
• Create the problem. Ideally, this will be a real-world situation that resembles something students may encounter in their future careers or lives. Cases are often the basis of PBL activities. Previously developed PBL activities can be found online through the University of Delaware’s PBL Clearinghouse of Activities .
• Establish ground rules at the beginning to prepare students to work effectively in groups.
• Introduce students to group processes and do some warm up exercises to allow them to practice assessing both their own work and that of their peers.
• Consider having students take on different roles or divide up the work up amongst themselves. Alternatively, the project might require students to assume various perspectives, such as those of government officials, local business owners, etc.
• Establish how you will evaluate and assess the assignment. Consider making the self and peer assessments a part of the assignment grade.

Nilson, L. B. (2010).  Teaching at its best: A research-based resource for college instructors  (2nd ed.).  San Francisco, CA: Jossey-Bass. 

Center for Teaching

Teaching problem solving.

Print Version

Tips and Techniques

Expert vs. novice problem solvers, communicate.

• Have students  identify specific problems, difficulties, or confusions . Don’t waste time working through problems that students already understand.
• If students are unable to articulate their concerns, determine where they are having trouble by  asking them to identify the specific concepts or principles associated with the problem.
• In a one-on-one tutoring session, ask the student to  work his/her problem out loud . This slows down the thinking process, making it more accurate and allowing you to access understanding.
• When working with larger groups you can ask students to provide a written “two-column solution.” Have students write up their solution to a problem by putting all their calculations in one column and all of their reasoning (in complete sentences) in the other column. This helps them to think critically about their own problem solving and helps you to more easily identify where they may be having problems. Two-Column Solution (Math) Two-Column Solution (Physics)

Encourage Independence

• Model the problem solving process rather than just giving students the answer. As you work through the problem, consider how a novice might struggle with the concepts and make your thinking clear
• Have students work through problems on their own. Ask directing questions or give helpful suggestions, but  provide only minimal assistance and only when needed to overcome obstacles.
• Don’t fear  group work ! Students can frequently help each other, and talking about a problem helps them think more critically about the steps needed to solve the problem. Additionally, group work helps students realize that problems often have multiple solution strategies, some that might be more effective than others

Be sensitive

• Frequently, when working problems, students are unsure of themselves. This lack of confidence may hamper their learning. It is important to recognize this when students come to us for help, and to give each student some feeling of mastery. Do this by providing  positive reinforcement to let students know when they have mastered a new concept or skill.

Encourage Thoroughness and Patience

• Try to communicate that  the process is more important than the answer so that the student learns that it is OK to not have an instant solution. This is learned through your acceptance of his/her pace of doing things, through your refusal to let anxiety pressure you into giving the right answer, and through your example of problem solving through a step-by step process.

Experts (teachers) in a particular field are often so fluent in solving problems from that field that they can find it difficult to articulate the problem solving principles and strategies they use to novices (students) in their field because these principles and strategies are second nature to the expert. To teach students problem solving skills,  a teacher should be aware of principles and strategies of good problem solving in his or her discipline .

The mathematician George Polya captured the problem solving principles and strategies he used in his discipline in the book  How to Solve It: A New Aspect of Mathematical Method (Princeton University Press, 1957). The book includes  a summary of Polya’s problem solving heuristic as well as advice on the teaching of problem solving.

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Problem-based learning (PBL): It’s all the rage. PBL is an innovative addition to modern K-12 pedagogy, but it can seem overwhelming for beginners. It doesn’t have to be, though.

PBL’s principal goal—meeting students where they are and putting them to work solving real-world problems—marks a significant shift from old educational models. Putting it into action unearths a host of practical challenges, from finding “real” real-world questions to ensuring accurate and adequate assessments.

1. Finding authentic problems

It can be difficult to know the kinds of problems that work best with PBL, but one thing’s for sure: Instructors must avoid manufacturing false challenges just to make them fit a predesigned pedagogy. Students are quick to identify these manufactured issues, said Dr. Richard Charles, the STEM and Innovation office director of Cherry Creek schools in Denver, Colorado, in a recent GlobalMindED panel. Charles called authenticity a key quality in PBL.

But what does “authentic” look like? An article on PBL at Education World recommends assigning students to ask local business leaders, other groups doing PBL, or key members of the school or local community to suggest tangible real-world problems. Projects might include:

• Investigating and addressing causes of student absences.
• Increasing a business’s website traffic.
• Helping a community develop a historic walking tour.

Whatever you choose, make sure it’s a real problem and that students target their solutions to a specific audience who is willing to listen. Furthermore, it’s still essential that PBL link with your school’s curriculum and standards.

Having students gather potential problems themselves offers a key opportunity to increase their metacognitive awareness of course outcomes or standards. They can gather their problems and then defend them as applicable to curriculum by illustrating potential areas of learning in their project.

These discussions help ground PBL in pedagogical soil—ensuring that students understand that PBL is both personally invented and defined, but also deeply educational. It’s all about real problems, real solutions and real learning.

2. Confronting your own lack of knowledge

Whether we intend to or not, instructors grow accustomed to “sage” status. PBL can move us off the stage and make us uncomfortable because we might not know or understand the problems or technologies our students are dealing with. That’s a unique challenge to our preparations, because we can walk with our students into areas that expose our ignorance.

One significant challenge to preparing for PBL is ensuring students have proper research resources and are prepared to do their own research and connect with mentors, educator Brianne Gidcumb suggests . While it may be impossible to fully prepare for a PBL journey with students, Gidcumb says that sincere problems that are justifiably connected to curricular outcomes can make teachers more comfortable with the process. Sometimes it is enough for us to understand that the journey is connected to our curriculum.

3. Getting up to speed on PBL

It is also important to be educated on PBL. While it is becoming increasingly popular, not every district has a wealth of professional development resources to ensure that teachers are deeply educated in PBL. This should not stop you. In addition to collecting your own resources, consider alternative methods of professional development like Twitter chats, message boards, or online communities dedicated to PBL.

PBL doesn’t have to be lonely territory for instructors. You can explore the creation of instructional teams or other in-school connections. PBL provides opportunities for work across disciplines, which gives teachers a chance to unite in planning. Having a small cohort to discuss instructional challenges is essential.

4. Embracing effective failure

The potential for failure is not a downside of PBL. Indeed, failure presents an excellent opportunity to show students their actions have real-world consequences and that success is far from guaranteed.

Students won’t be able to design clear solutions to every problem they encounter. Industries understand that failure is sometimes a piece of the process; schools should have the same approach. Students need to understand that failure is not only accepted out there in the real world: It’s often a key part of the process.

Communicating the PBL process to students is key. They must see PBL as an opportunity to unite real-world issues with academic outcomes and standards. They must understand PBL isn’t an exam and perfection is not the goal.

Instructor Carmel Schettino identifies communication as a key challenge to PBL. In a blog post for the National Council of Teachers of Mathematics, she identifies two key pieces of the assessment process for PBL:

• Instructors must consistently engage students to help them understand how their PBL work links back to their overall learning outcomes or course objectives.
• Students need opportunities to revise in their work so they can reflect on how PBL relates to their classroom assessments.

There are all sorts of ways to assess students’ success on PBL. Whatever you do, make sure you’re communicating in a way that ensures they understand the applicability of the work they are doing.

Though wading into unfamiliar teaching territory can be scary, keep in mind that problem-based learning is not new. It has been consistently used since the 1960s in medical schools. Adoption at the K-12 level can seem daunting, but the skills it imparts on students, both in connecting with curricular outcomes and in soft-skills like communication, frustration and effective failure, are essential in 21st century learners. Embrace PBL with verve and see every new challenge as ab opportunity for significant and measurable learning.

You may also like to read

• Three Websites For Project-Based Learning
• Explanation of Brain Based Learning
• Teaching Algebra Using Project-Based Learning
• An Introduction to Project-Based Learning
• Will Brain-Based Learning Help Your Students?
• Get Your Students More Involved With Project-Based Learning

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The advantages and disadvantages of problem-solving practice when learning basic addition facts

Research output : Chapter in Book/Report/Conference proceeding › Chapter (Book) › Research › peer-review

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• 10.4324/9780429400902-15

T1 - The advantages and disadvantages of problem-solving practice when learning basic addition facts

AU - Hopkins, Sarah

N2 - How children learn to retrieve answers to basic (single-digit) addition problems and how teachers can support children's learning of retrieval has captivated my attention as a researcher and teacher educator for the last 25 years. In this chapter, I describe this research and explain how I got started with the help of Professor Mike Lawson. I then present findings from a series of microgenetic studies to illustrate the different effects problem-solving practice has on children's development of retrieval.

AB - How children learn to retrieve answers to basic (single-digit) addition problems and how teachers can support children's learning of retrieval has captivated my attention as a researcher and teacher educator for the last 25 years. In this chapter, I describe this research and explain how I got started with the help of Professor Mike Lawson. I then present findings from a series of microgenetic studies to illustrate the different effects problem-solving practice has on children's development of retrieval.

U2 - 10.4324/9780429400902-15

DO - 10.4324/9780429400902-15

M3 - Chapter (Book)

SN - 9780367001834

BT - Problem Solving for Teaching and Learning

A2 - Askell-Williams, Helen

A2 - Orrell, Janice

PB - Routledge

CY - Abingdon UK

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Problem-Based Learning (PBL) is a teaching method in which complex real-world problems are used as the vehicle to promote student learning of concepts and principles as opposed to direct presentation of facts and concepts. In addition to course content, PBL can promote the development of critical thinking skills, problem-solving abilities, and communication skills. It can also provide opportunities for working in groups, finding and evaluating research materials, and life-long learning (Duch et al, 2001).

PBL can be incorporated into any learning situation. In the strictest definition of PBL, the approach is used over the entire semester as the primary method of teaching. However, broader definitions and uses range from including PBL in lab and design classes, to using it simply to start a single discussion. PBL can also be used to create assessment items. The main thread connecting these various uses is the real-world problem.

Any subject area can be adapted to PBL with a little creativity. While the core problems will vary among disciplines, there are some characteristics of good PBL problems that transcend fields (Duch, Groh, and Allen, 2001):

• The problem must motivate students to seek out a deeper understanding of concepts.
• The problem should require students to make reasoned decisions and to defend them.
• The problem should incorporate the content objectives in such a way as to connect it to previous courses/knowledge.
• If used for a group project, the problem needs a level of complexity to ensure that the students must work together to solve it.
• If used for a multistage project, the initial steps of the problem should be open-ended and engaging to draw students into the problem.

The problems can come from a variety of sources: newspapers, magazines, journals, books, textbooks, and television/ movies. Some are in such form that they can be used with little editing; however, others need to be rewritten to be of use. The following guidelines from The Power of Problem-Based Learning (Duch et al, 2001) are written for creating PBL problems for a class centered around the method; however, the general ideas can be applied in simpler uses of PBL:

• Choose a central idea, concept, or principle that is always taught in a given course, and then think of a typical end-of-chapter problem, assignment, or homework that is usually assigned to students to help them learn that concept. List the learning objectives that students should meet when they work through the problem.
• Think of a real-world context for the concept under consideration. Develop a storytelling aspect to an end-of-chapter problem, or research an actual case that can be adapted, adding some motivation for students to solve the problem. More complex problems will challenge students to go beyond simple plug-and-chug to solve it. Look at magazines, newspapers, and articles for ideas on the story line. Some PBL practitioners talk to professionals in the field, searching for ideas of realistic applications of the concept being taught.
• What will the first page (or stage) look like? What open-ended questions can be asked? What learning issues will be identified?
• How will the problem be structured?
• How long will the problem be? How many class periods will it take to complete?
• Will students be given information in subsequent pages (or stages) as they work through the problem?
• What resources will the students need?
• What end product will the students produce at the completion of the problem?
• Write a teacher's guide detailing the instructional plans on using the problem in the course. If the course is a medium- to large-size class, a combination of mini-lectures, whole-class discussions, and small group work with regular reporting may be necessary. The teacher's guide can indicate plans or options for cycling through the pages of the problem interspersing the various modes of learning.
• The final step is to identify key resources for students. Students need to learn to identify and utilize learning resources on their own, but it can be helpful if the instructor indicates a few good sources to get them started. Many students will want to limit their research to the Internet, so it will be important to guide them toward the library as well.

The method for distributing a PBL problem falls under three closely related teaching techniques: case studies, role-plays, and simulations. Case studies are presented to students in written form. Role-plays have students improvise scenes based on character descriptions given. Today, simulations often involve computer-based programs. Regardless of which technique is used, the heart of the method remains the same: the real-world problem.

Where can I learn more?

• PBL through the Institute for Transforming Undergraduate Education at the University of Delaware
• Duch, B. J., Groh, S. E, & Allen, D. E. (Eds.). (2001). The power of problem-based learning . Sterling, VA: Stylus.
• Grasha, A. F. (1996). Teaching with style: A practical guide to enhancing learning by understanding teaching and learning styles. Pittsburgh: Alliance Publishers.

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Problem-Solving Method in Teaching

The problem-solving method is a highly effective teaching strategy that is designed to help students develop critical thinking skills and problem-solving abilities . It involves providing students with real-world problems and challenges that require them to apply their knowledge, skills, and creativity to find solutions. This method encourages active learning, promotes collaboration, and allows students to take ownership of their learning.

Table of Contents

Definition of problem-solving method.

Problem-solving is a process of identifying, analyzing, and resolving problems. The problem-solving method in teaching involves providing students with real-world problems that they must solve through collaboration and critical thinking. This method encourages students to apply their knowledge and creativity to develop solutions that are effective and practical.

Meaning of Problem-Solving Method

The meaning and Definition of problem-solving are given by different Scholars. These are-

Woodworth and Marquis(1948) : Problem-solving behavior occurs in novel or difficult situations in which a solution is not obtainable by the habitual methods of applying concepts and principles derived from past experience in very similar situations.

Skinner (1968): Problem-solving is a process of overcoming difficulties that appear to interfere with the attainment of a goal. It is the procedure of making adjustments in spite of interference

Benefits of Problem-Solving Method

The problem-solving method has several benefits for both students and teachers. These benefits include:

• Encourages active learning: The problem-solving method encourages students to actively participate in their own learning by engaging them in real-world problems that require critical thinking and collaboration
• Promotes collaboration: Problem-solving requires students to work together to find solutions. This promotes teamwork, communication, and cooperation.
• Builds critical thinking skills: The problem-solving method helps students develop critical thinking skills by providing them with opportunities to analyze and evaluate problems
• Increases motivation: When students are engaged in solving real-world problems, they are more motivated to learn and apply their knowledge.
• Enhances creativity: The problem-solving method encourages students to be creative in finding solutions to problems.

Steps in Problem-Solving Method

The problem-solving method involves several steps that teachers can use to guide their students. These steps include

• Identifying the problem: The first step in problem-solving is identifying the problem that needs to be solved. Teachers can present students with a real-world problem or challenge that requires critical thinking and collaboration.
• Analyzing the problem: Once the problem is identified, students should analyze it to determine its scope and underlying causes.
• Generating solutions: After analyzing the problem, students should generate possible solutions. This step requires creativity and critical thinking.
• Evaluating solutions: The next step is to evaluate each solution based on its effectiveness and practicality
• Selecting the best solution: The final step is to select the best solution and implement it.

Verification of the concluded solution or Hypothesis

The solution arrived at or the conclusion drawn must be further verified by utilizing it in solving various other likewise problems. In case, the derived solution helps in solving these problems, then and only then if one is free to agree with his finding regarding the solution. The verified solution may then become a useful product of his problem-solving behavior that can be utilized in solving further problems. The above steps can be utilized in solving various problems thereby fostering creative thinking ability in an individual.

The problem-solving method is an effective teaching strategy that promotes critical thinking, creativity, and collaboration. It provides students with real-world problems that require them to apply their knowledge and skills to find solutions. By using the problem-solving method, teachers can help their students develop the skills they need to succeed in school and in life.

• Jonassen, D. (2011). Learning to solve problems: A handbook for designing problem-solving learning environments. Routledge.
• Hmelo-Silver, C. E. (2004). Problem-based learning: What and how do students learn? Educational Psychology Review, 16(3), 235-266.
• Mergendoller, J. R., Maxwell, N. L., & Bellisimo, Y. (2006). The effectiveness of problem-based instruction: A comparative study of instructional methods and student characteristics. Interdisciplinary Journal of Problem-based Learning, 1(2), 49-69.
• Richey, R. C., Klein, J. D., & Tracey, M. W. (2011). The instructional design knowledge base: Theory, research, and practice. Routledge.
• Savery, J. R., & Duffy, T. M. (2001). Problem-based learning: An instructional model and its constructivist framework. CRLT Technical Report No. 16-01, University of Michigan. Wojcikowski, J. (2013). Solving real-world problems through problem-based learning. College Teaching, 61(4), 153-156

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• Published: 11 January 2023

The effectiveness of collaborative problem solving in promoting students’ critical thinking: A meta-analysis based on empirical literature

• Enwei Xu   ORCID: orcid.org/0000-0001-6424-8169 1 ,
• Wei Wang 1 &
• Qingxia Wang 1  

Humanities and Social Sciences Communications volume  10 , Article number:  16 ( 2023 ) Cite this article

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Collaborative problem-solving has been widely embraced in the classroom instruction of critical thinking, which is regarded as the core of curriculum reform based on key competencies in the field of education as well as a key competence for learners in the 21st century. However, the effectiveness of collaborative problem-solving in promoting students’ critical thinking remains uncertain. This current research presents the major findings of a meta-analysis of 36 pieces of the literature revealed in worldwide educational periodicals during the 21st century to identify the effectiveness of collaborative problem-solving in promoting students’ critical thinking and to determine, based on evidence, whether and to what extent collaborative problem solving can result in a rise or decrease in critical thinking. The findings show that (1) collaborative problem solving is an effective teaching approach to foster students’ critical thinking, with a significant overall effect size (ES = 0.82, z  = 12.78, P  < 0.01, 95% CI [0.69, 0.95]); (2) in respect to the dimensions of critical thinking, collaborative problem solving can significantly and successfully enhance students’ attitudinal tendencies (ES = 1.17, z  = 7.62, P  < 0.01, 95% CI[0.87, 1.47]); nevertheless, it falls short in terms of improving students’ cognitive skills, having only an upper-middle impact (ES = 0.70, z  = 11.55, P  < 0.01, 95% CI[0.58, 0.82]); and (3) the teaching type (chi 2  = 7.20, P  < 0.05), intervention duration (chi 2  = 12.18, P  < 0.01), subject area (chi 2  = 13.36, P  < 0.05), group size (chi 2  = 8.77, P  < 0.05), and learning scaffold (chi 2  = 9.03, P  < 0.01) all have an impact on critical thinking, and they can be viewed as important moderating factors that affect how critical thinking develops. On the basis of these results, recommendations are made for further study and instruction to better support students’ critical thinking in the context of collaborative problem-solving.

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Introduction.

Although critical thinking has a long history in research, the concept of critical thinking, which is regarded as an essential competence for learners in the 21st century, has recently attracted more attention from researchers and teaching practitioners (National Research Council, 2012 ). Critical thinking should be the core of curriculum reform based on key competencies in the field of education (Peng and Deng, 2017 ) because students with critical thinking can not only understand the meaning of knowledge but also effectively solve practical problems in real life even after knowledge is forgotten (Kek and Huijser, 2011 ). The definition of critical thinking is not universal (Ennis, 1989 ; Castle, 2009 ; Niu et al., 2013 ). In general, the definition of critical thinking is a self-aware and self-regulated thought process (Facione, 1990 ; Niu et al., 2013 ). It refers to the cognitive skills needed to interpret, analyze, synthesize, reason, and evaluate information as well as the attitudinal tendency to apply these abilities (Halpern, 2001 ). The view that critical thinking can be taught and learned through curriculum teaching has been widely supported by many researchers (e.g., Kuncel, 2011 ; Leng and Lu, 2020 ), leading to educators’ efforts to foster it among students. In the field of teaching practice, there are three types of courses for teaching critical thinking (Ennis, 1989 ). The first is an independent curriculum in which critical thinking is taught and cultivated without involving the knowledge of specific disciplines; the second is an integrated curriculum in which critical thinking is integrated into the teaching of other disciplines as a clear teaching goal; and the third is a mixed curriculum in which critical thinking is taught in parallel to the teaching of other disciplines for mixed teaching training. Furthermore, numerous measuring tools have been developed by researchers and educators to measure critical thinking in the context of teaching practice. These include standardized measurement tools, such as WGCTA, CCTST, CCTT, and CCTDI, which have been verified by repeated experiments and are considered effective and reliable by international scholars (Facione and Facione, 1992 ). In short, descriptions of critical thinking, including its two dimensions of attitudinal tendency and cognitive skills, different types of teaching courses, and standardized measurement tools provide a complex normative framework for understanding, teaching, and evaluating critical thinking.

Cultivating critical thinking in curriculum teaching can start with a problem, and one of the most popular critical thinking instructional approaches is problem-based learning (Liu et al., 2020 ). Duch et al. ( 2001 ) noted that problem-based learning in group collaboration is progressive active learning, which can improve students’ critical thinking and problem-solving skills. Collaborative problem-solving is the organic integration of collaborative learning and problem-based learning, which takes learners as the center of the learning process and uses problems with poor structure in real-world situations as the starting point for the learning process (Liang et al., 2017 ). Students learn the knowledge needed to solve problems in a collaborative group, reach a consensus on problems in the field, and form solutions through social cooperation methods, such as dialogue, interpretation, questioning, debate, negotiation, and reflection, thus promoting the development of learners’ domain knowledge and critical thinking (Cindy, 2004 ; Liang et al., 2017 ).

Collaborative problem-solving has been widely used in the teaching practice of critical thinking, and several studies have attempted to conduct a systematic review and meta-analysis of the empirical literature on critical thinking from various perspectives. However, little attention has been paid to the impact of collaborative problem-solving on critical thinking. Therefore, the best approach for developing and enhancing critical thinking throughout collaborative problem-solving is to examine how to implement critical thinking instruction; however, this issue is still unexplored, which means that many teachers are incapable of better instructing critical thinking (Leng and Lu, 2020 ; Niu et al., 2013 ). For example, Huber ( 2016 ) provided the meta-analysis findings of 71 publications on gaining critical thinking over various time frames in college with the aim of determining whether critical thinking was truly teachable. These authors found that learners significantly improve their critical thinking while in college and that critical thinking differs with factors such as teaching strategies, intervention duration, subject area, and teaching type. The usefulness of collaborative problem-solving in fostering students’ critical thinking, however, was not determined by this study, nor did it reveal whether there existed significant variations among the different elements. A meta-analysis of 31 pieces of educational literature was conducted by Liu et al. ( 2020 ) to assess the impact of problem-solving on college students’ critical thinking. These authors found that problem-solving could promote the development of critical thinking among college students and proposed establishing a reasonable group structure for problem-solving in a follow-up study to improve students’ critical thinking. Additionally, previous empirical studies have reached inconclusive and even contradictory conclusions about whether and to what extent collaborative problem-solving increases or decreases critical thinking levels. As an illustration, Yang et al. ( 2008 ) carried out an experiment on the integrated curriculum teaching of college students based on a web bulletin board with the goal of fostering participants’ critical thinking in the context of collaborative problem-solving. These authors’ research revealed that through sharing, debating, examining, and reflecting on various experiences and ideas, collaborative problem-solving can considerably enhance students’ critical thinking in real-life problem situations. In contrast, collaborative problem-solving had a positive impact on learners’ interaction and could improve learning interest and motivation but could not significantly improve students’ critical thinking when compared to traditional classroom teaching, according to research by Naber and Wyatt ( 2014 ) and Sendag and Odabasi ( 2009 ) on undergraduate and high school students, respectively.

The above studies show that there is inconsistency regarding the effectiveness of collaborative problem-solving in promoting students’ critical thinking. Therefore, it is essential to conduct a thorough and trustworthy review to detect and decide whether and to what degree collaborative problem-solving can result in a rise or decrease in critical thinking. Meta-analysis is a quantitative analysis approach that is utilized to examine quantitative data from various separate studies that are all focused on the same research topic. This approach characterizes the effectiveness of its impact by averaging the effect sizes of numerous qualitative studies in an effort to reduce the uncertainty brought on by independent research and produce more conclusive findings (Lipsey and Wilson, 2001 ).

This paper used a meta-analytic approach and carried out a meta-analysis to examine the effectiveness of collaborative problem-solving in promoting students’ critical thinking in order to make a contribution to both research and practice. The following research questions were addressed by this meta-analysis:

What is the overall effect size of collaborative problem-solving in promoting students’ critical thinking and its impact on the two dimensions of critical thinking (i.e., attitudinal tendency and cognitive skills)?

How are the disparities between the study conclusions impacted by various moderating variables if the impacts of various experimental designs in the included studies are heterogeneous?

This research followed the strict procedures (e.g., database searching, identification, screening, eligibility, merging, duplicate removal, and analysis of included studies) of Cooper’s ( 2010 ) proposed meta-analysis approach for examining quantitative data from various separate studies that are all focused on the same research topic. The relevant empirical research that appeared in worldwide educational periodicals within the 21st century was subjected to this meta-analysis using Rev-Man 5.4. The consistency of the data extracted separately by two researchers was tested using Cohen’s kappa coefficient, and a publication bias test and a heterogeneity test were run on the sample data to ascertain the quality of this meta-analysis.

Data sources and search strategies

There were three stages to the data collection process for this meta-analysis, as shown in Fig. 1 , which shows the number of articles included and eliminated during the selection process based on the statement and study eligibility criteria.

This flowchart shows the number of records identified, included and excluded in the article.

First, the databases used to systematically search for relevant articles were the journal papers of the Web of Science Core Collection and the Chinese Core source journal, as well as the Chinese Social Science Citation Index (CSSCI) source journal papers included in CNKI. These databases were selected because they are credible platforms that are sources of scholarly and peer-reviewed information with advanced search tools and contain literature relevant to the subject of our topic from reliable researchers and experts. The search string with the Boolean operator used in the Web of Science was “TS = (((“critical thinking” or “ct” and “pretest” or “posttest”) or (“critical thinking” or “ct” and “control group” or “quasi experiment” or “experiment”)) and (“collaboration” or “collaborative learning” or “CSCL”) and (“problem solving” or “problem-based learning” or “PBL”))”. The research area was “Education Educational Research”, and the search period was “January 1, 2000, to December 30, 2021”. A total of 412 papers were obtained. The search string with the Boolean operator used in the CNKI was “SU = (‘critical thinking’*‘collaboration’ + ‘critical thinking’*‘collaborative learning’ + ‘critical thinking’*‘CSCL’ + ‘critical thinking’*‘problem solving’ + ‘critical thinking’*‘problem-based learning’ + ‘critical thinking’*‘PBL’ + ‘critical thinking’*‘problem oriented’) AND FT = (‘experiment’ + ‘quasi experiment’ + ‘pretest’ + ‘posttest’ + ‘empirical study’)” (translated into Chinese when searching). A total of 56 studies were found throughout the search period of “January 2000 to December 2021”. From the databases, all duplicates and retractions were eliminated before exporting the references into Endnote, a program for managing bibliographic references. In all, 466 studies were found.

Second, the studies that matched the inclusion and exclusion criteria for the meta-analysis were chosen by two researchers after they had reviewed the abstracts and titles of the gathered articles, yielding a total of 126 studies.

Third, two researchers thoroughly reviewed each included article’s whole text in accordance with the inclusion and exclusion criteria. Meanwhile, a snowball search was performed using the references and citations of the included articles to ensure complete coverage of the articles. Ultimately, 36 articles were kept.

Two researchers worked together to carry out this entire process, and a consensus rate of almost 94.7% was reached after discussion and negotiation to clarify any emerging differences.

Eligibility criteria

Since not all the retrieved studies matched the criteria for this meta-analysis, eligibility criteria for both inclusion and exclusion were developed as follows:

The publication language of the included studies was limited to English and Chinese, and the full text could be obtained. Articles that did not meet the publication language and articles not published between 2000 and 2021 were excluded.

The research design of the included studies must be empirical and quantitative studies that can assess the effect of collaborative problem-solving on the development of critical thinking. Articles that could not identify the causal mechanisms by which collaborative problem-solving affects critical thinking, such as review articles and theoretical articles, were excluded.

The research method of the included studies must feature a randomized control experiment or a quasi-experiment, or a natural experiment, which have a higher degree of internal validity with strong experimental designs and can all plausibly provide evidence that critical thinking and collaborative problem-solving are causally related. Articles with non-experimental research methods, such as purely correlational or observational studies, were excluded.

The participants of the included studies were only students in school, including K-12 students and college students. Articles in which the participants were non-school students, such as social workers or adult learners, were excluded.

The research results of the included studies must mention definite signs that may be utilized to gauge critical thinking’s impact (e.g., sample size, mean value, or standard deviation). Articles that lacked specific measurement indicators for critical thinking and could not calculate the effect size were excluded.

Data coding design

In order to perform a meta-analysis, it is necessary to collect the most important information from the articles, codify that information’s properties, and convert descriptive data into quantitative data. Therefore, this study designed a data coding template (see Table 1 ). Ultimately, 16 coding fields were retained.

The designed data-coding template consisted of three pieces of information. Basic information about the papers was included in the descriptive information: the publishing year, author, serial number, and title of the paper.

The variable information for the experimental design had three variables: the independent variable (instruction method), the dependent variable (critical thinking), and the moderating variable (learning stage, teaching type, intervention duration, learning scaffold, group size, measuring tool, and subject area). Depending on the topic of this study, the intervention strategy, as the independent variable, was coded into collaborative and non-collaborative problem-solving. The dependent variable, critical thinking, was coded as a cognitive skill and an attitudinal tendency. And seven moderating variables were created by grouping and combining the experimental design variables discovered within the 36 studies (see Table 1 ), where learning stages were encoded as higher education, high school, middle school, and primary school or lower; teaching types were encoded as mixed courses, integrated courses, and independent courses; intervention durations were encoded as 0–1 weeks, 1–4 weeks, 4–12 weeks, and more than 12 weeks; group sizes were encoded as 2–3 persons, 4–6 persons, 7–10 persons, and more than 10 persons; learning scaffolds were encoded as teacher-supported learning scaffold, technique-supported learning scaffold, and resource-supported learning scaffold; measuring tools were encoded as standardized measurement tools (e.g., WGCTA, CCTT, CCTST, and CCTDI) and self-adapting measurement tools (e.g., modified or made by researchers); and subject areas were encoded according to the specific subjects used in the 36 included studies.

The data information contained three metrics for measuring critical thinking: sample size, average value, and standard deviation. It is vital to remember that studies with various experimental designs frequently adopt various formulas to determine the effect size. And this paper used Morris’ proposed standardized mean difference (SMD) calculation formula ( 2008 , p. 369; see Supplementary Table S3 ).

Procedure for extracting and coding data

According to the data coding template (see Table 1 ), the 36 papers’ information was retrieved by two researchers, who then entered them into Excel (see Supplementary Table S1 ). The results of each study were extracted separately in the data extraction procedure if an article contained numerous studies on critical thinking, or if a study assessed different critical thinking dimensions. For instance, Tiwari et al. ( 2010 ) used four time points, which were viewed as numerous different studies, to examine the outcomes of critical thinking, and Chen ( 2013 ) included the two outcome variables of attitudinal tendency and cognitive skills, which were regarded as two studies. After discussion and negotiation during data extraction, the two researchers’ consistency test coefficients were roughly 93.27%. Supplementary Table S2 details the key characteristics of the 36 included articles with 79 effect quantities, including descriptive information (e.g., the publishing year, author, serial number, and title of the paper), variable information (e.g., independent variables, dependent variables, and moderating variables), and data information (e.g., mean values, standard deviations, and sample size). Following that, testing for publication bias and heterogeneity was done on the sample data using the Rev-Man 5.4 software, and then the test results were used to conduct a meta-analysis.

Publication bias test

When the sample of studies included in a meta-analysis does not accurately reflect the general status of research on the relevant subject, publication bias is said to be exhibited in this research. The reliability and accuracy of the meta-analysis may be impacted by publication bias. Due to this, the meta-analysis needs to check the sample data for publication bias (Stewart et al., 2006 ). A popular method to check for publication bias is the funnel plot; and it is unlikely that there will be publishing bias when the data are equally dispersed on either side of the average effect size and targeted within the higher region. The data are equally dispersed within the higher portion of the efficient zone, consistent with the funnel plot connected with this analysis (see Fig. 2 ), indicating that publication bias is unlikely in this situation.

This funnel plot shows the result of publication bias of 79 effect quantities across 36 studies.

Heterogeneity test

To select the appropriate effect models for the meta-analysis, one might use the results of a heterogeneity test on the data effect sizes. In a meta-analysis, it is common practice to gauge the degree of data heterogeneity using the I 2 value, and I 2  ≥ 50% is typically understood to denote medium-high heterogeneity, which calls for the adoption of a random effect model; if not, a fixed effect model ought to be applied (Lipsey and Wilson, 2001 ). The findings of the heterogeneity test in this paper (see Table 2 ) revealed that I 2 was 86% and displayed significant heterogeneity ( P  < 0.01). To ensure accuracy and reliability, the overall effect size ought to be calculated utilizing the random effect model.

The analysis of the overall effect size

This meta-analysis utilized a random effect model to examine 79 effect quantities from 36 studies after eliminating heterogeneity. In accordance with Cohen’s criterion (Cohen, 1992 ), it is abundantly clear from the analysis results, which are shown in the forest plot of the overall effect (see Fig. 3 ), that the cumulative impact size of cooperative problem-solving is 0.82, which is statistically significant ( z  = 12.78, P  < 0.01, 95% CI [0.69, 0.95]), and can encourage learners to practice critical thinking.

This forest plot shows the analysis result of the overall effect size across 36 studies.

In addition, this study examined two distinct dimensions of critical thinking to better understand the precise contributions that collaborative problem-solving makes to the growth of critical thinking. The findings (see Table 3 ) indicate that collaborative problem-solving improves cognitive skills (ES = 0.70) and attitudinal tendency (ES = 1.17), with significant intergroup differences (chi 2  = 7.95, P  < 0.01). Although collaborative problem-solving improves both dimensions of critical thinking, it is essential to point out that the improvements in students’ attitudinal tendency are much more pronounced and have a significant comprehensive effect (ES = 1.17, z  = 7.62, P  < 0.01, 95% CI [0.87, 1.47]), whereas gains in learners’ cognitive skill are slightly improved and are just above average. (ES = 0.70, z  = 11.55, P  < 0.01, 95% CI [0.58, 0.82]).

The analysis of moderator effect size

The whole forest plot’s 79 effect quantities underwent a two-tailed test, which revealed significant heterogeneity ( I 2  = 86%, z  = 12.78, P  < 0.01), indicating differences between various effect sizes that may have been influenced by moderating factors other than sampling error. Therefore, exploring possible moderating factors that might produce considerable heterogeneity was done using subgroup analysis, such as the learning stage, learning scaffold, teaching type, group size, duration of the intervention, measuring tool, and the subject area included in the 36 experimental designs, in order to further explore the key factors that influence critical thinking. The findings (see Table 4 ) indicate that various moderating factors have advantageous effects on critical thinking. In this situation, the subject area (chi 2  = 13.36, P  < 0.05), group size (chi 2  = 8.77, P  < 0.05), intervention duration (chi 2  = 12.18, P  < 0.01), learning scaffold (chi 2  = 9.03, P  < 0.01), and teaching type (chi 2  = 7.20, P  < 0.05) are all significant moderators that can be applied to support the cultivation of critical thinking. However, since the learning stage and the measuring tools did not significantly differ among intergroup (chi 2  = 3.15, P  = 0.21 > 0.05, and chi 2  = 0.08, P  = 0.78 > 0.05), we are unable to explain why these two factors are crucial in supporting the cultivation of critical thinking in the context of collaborative problem-solving. These are the precise outcomes, as follows:

Various learning stages influenced critical thinking positively, without significant intergroup differences (chi 2  = 3.15, P  = 0.21 > 0.05). High school was first on the list of effect sizes (ES = 1.36, P  < 0.01), then higher education (ES = 0.78, P  < 0.01), and middle school (ES = 0.73, P  < 0.01). These results show that, despite the learning stage’s beneficial influence on cultivating learners’ critical thinking, we are unable to explain why it is essential for cultivating critical thinking in the context of collaborative problem-solving.

Different teaching types had varying degrees of positive impact on critical thinking, with significant intergroup differences (chi 2  = 7.20, P  < 0.05). The effect size was ranked as follows: mixed courses (ES = 1.34, P  < 0.01), integrated courses (ES = 0.81, P  < 0.01), and independent courses (ES = 0.27, P  < 0.01). These results indicate that the most effective approach to cultivate critical thinking utilizing collaborative problem solving is through the teaching type of mixed courses.

Various intervention durations significantly improved critical thinking, and there were significant intergroup differences (chi 2  = 12.18, P  < 0.01). The effect sizes related to this variable showed a tendency to increase with longer intervention durations. The improvement in critical thinking reached a significant level (ES = 0.85, P  < 0.01) after more than 12 weeks of training. These findings indicate that the intervention duration and critical thinking’s impact are positively correlated, with a longer intervention duration having a greater effect.

Different learning scaffolds influenced critical thinking positively, with significant intergroup differences (chi 2  = 9.03, P  < 0.01). The resource-supported learning scaffold (ES = 0.69, P  < 0.01) acquired a medium-to-higher level of impact, the technique-supported learning scaffold (ES = 0.63, P  < 0.01) also attained a medium-to-higher level of impact, and the teacher-supported learning scaffold (ES = 0.92, P  < 0.01) displayed a high level of significant impact. These results show that the learning scaffold with teacher support has the greatest impact on cultivating critical thinking.

Various group sizes influenced critical thinking positively, and the intergroup differences were statistically significant (chi 2  = 8.77, P  < 0.05). Critical thinking showed a general declining trend with increasing group size. The overall effect size of 2–3 people in this situation was the biggest (ES = 0.99, P  < 0.01), and when the group size was greater than 7 people, the improvement in critical thinking was at the lower-middle level (ES < 0.5, P  < 0.01). These results show that the impact on critical thinking is positively connected with group size, and as group size grows, so does the overall impact.

Various measuring tools influenced critical thinking positively, with significant intergroup differences (chi 2  = 0.08, P  = 0.78 > 0.05). In this situation, the self-adapting measurement tools obtained an upper-medium level of effect (ES = 0.78), whereas the complete effect size of the standardized measurement tools was the largest, achieving a significant level of effect (ES = 0.84, P  < 0.01). These results show that, despite the beneficial influence of the measuring tool on cultivating critical thinking, we are unable to explain why it is crucial in fostering the growth of critical thinking by utilizing the approach of collaborative problem-solving.

Different subject areas had a greater impact on critical thinking, and the intergroup differences were statistically significant (chi 2  = 13.36, P  < 0.05). Mathematics had the greatest overall impact, achieving a significant level of effect (ES = 1.68, P  < 0.01), followed by science (ES = 1.25, P  < 0.01) and medical science (ES = 0.87, P  < 0.01), both of which also achieved a significant level of effect. Programming technology was the least effective (ES = 0.39, P  < 0.01), only having a medium-low degree of effect compared to education (ES = 0.72, P  < 0.01) and other fields (such as language, art, and social sciences) (ES = 0.58, P  < 0.01). These results suggest that scientific fields (e.g., mathematics, science) may be the most effective subject areas for cultivating critical thinking utilizing the approach of collaborative problem-solving.

The effectiveness of collaborative problem solving with regard to teaching critical thinking

According to this meta-analysis, using collaborative problem-solving as an intervention strategy in critical thinking teaching has a considerable amount of impact on cultivating learners’ critical thinking as a whole and has a favorable promotional effect on the two dimensions of critical thinking. According to certain studies, collaborative problem solving, the most frequently used critical thinking teaching strategy in curriculum instruction can considerably enhance students’ critical thinking (e.g., Liang et al., 2017 ; Liu et al., 2020 ; Cindy, 2004 ). This meta-analysis provides convergent data support for the above research views. Thus, the findings of this meta-analysis not only effectively address the first research query regarding the overall effect of cultivating critical thinking and its impact on the two dimensions of critical thinking (i.e., attitudinal tendency and cognitive skills) utilizing the approach of collaborative problem-solving, but also enhance our confidence in cultivating critical thinking by using collaborative problem-solving intervention approach in the context of classroom teaching.

Furthermore, the associated improvements in attitudinal tendency are much stronger, but the corresponding improvements in cognitive skill are only marginally better. According to certain studies, cognitive skill differs from the attitudinal tendency in classroom instruction; the cultivation and development of the former as a key ability is a process of gradual accumulation, while the latter as an attitude is affected by the context of the teaching situation (e.g., a novel and exciting teaching approach, challenging and rewarding tasks) (Halpern, 2001 ; Wei and Hong, 2022 ). Collaborative problem-solving as a teaching approach is exciting and interesting, as well as rewarding and challenging; because it takes the learners as the focus and examines problems with poor structure in real situations, and it can inspire students to fully realize their potential for problem-solving, which will significantly improve their attitudinal tendency toward solving problems (Liu et al., 2020 ). Similar to how collaborative problem-solving influences attitudinal tendency, attitudinal tendency impacts cognitive skill when attempting to solve a problem (Liu et al., 2020 ; Zhang et al., 2022 ), and stronger attitudinal tendencies are associated with improved learning achievement and cognitive ability in students (Sison, 2008 ; Zhang et al., 2022 ). It can be seen that the two specific dimensions of critical thinking as well as critical thinking as a whole are affected by collaborative problem-solving, and this study illuminates the nuanced links between cognitive skills and attitudinal tendencies with regard to these two dimensions of critical thinking. To fully develop students’ capacity for critical thinking, future empirical research should pay closer attention to cognitive skills.

The moderating effects of collaborative problem solving with regard to teaching critical thinking

In order to further explore the key factors that influence critical thinking, exploring possible moderating effects that might produce considerable heterogeneity was done using subgroup analysis. The findings show that the moderating factors, such as the teaching type, learning stage, group size, learning scaffold, duration of the intervention, measuring tool, and the subject area included in the 36 experimental designs, could all support the cultivation of collaborative problem-solving in critical thinking. Among them, the effect size differences between the learning stage and measuring tool are not significant, which does not explain why these two factors are crucial in supporting the cultivation of critical thinking utilizing the approach of collaborative problem-solving.

In terms of the learning stage, various learning stages influenced critical thinking positively without significant intergroup differences, indicating that we are unable to explain why it is crucial in fostering the growth of critical thinking.

Although high education accounts for 70.89% of all empirical studies performed by researchers, high school may be the appropriate learning stage to foster students’ critical thinking by utilizing the approach of collaborative problem-solving since it has the largest overall effect size. This phenomenon may be related to student’s cognitive development, which needs to be further studied in follow-up research.

With regard to teaching type, mixed course teaching may be the best teaching method to cultivate students’ critical thinking. Relevant studies have shown that in the actual teaching process if students are trained in thinking methods alone, the methods they learn are isolated and divorced from subject knowledge, which is not conducive to their transfer of thinking methods; therefore, if students’ thinking is trained only in subject teaching without systematic method training, it is challenging to apply to real-world circumstances (Ruggiero, 2012 ; Hu and Liu, 2015 ). Teaching critical thinking as mixed course teaching in parallel to other subject teachings can achieve the best effect on learners’ critical thinking, and explicit critical thinking instruction is more effective than less explicit critical thinking instruction (Bensley and Spero, 2014 ).

In terms of the intervention duration, with longer intervention times, the overall effect size shows an upward tendency. Thus, the intervention duration and critical thinking’s impact are positively correlated. Critical thinking, as a key competency for students in the 21st century, is difficult to get a meaningful improvement in a brief intervention duration. Instead, it could be developed over a lengthy period of time through consistent teaching and the progressive accumulation of knowledge (Halpern, 2001 ; Hu and Liu, 2015 ). Therefore, future empirical studies ought to take these restrictions into account throughout a longer period of critical thinking instruction.

With regard to group size, a group size of 2–3 persons has the highest effect size, and the comprehensive effect size decreases with increasing group size in general. This outcome is in line with some research findings; as an example, a group composed of two to four members is most appropriate for collaborative learning (Schellens and Valcke, 2006 ). However, the meta-analysis results also indicate that once the group size exceeds 7 people, small groups cannot produce better interaction and performance than large groups. This may be because the learning scaffolds of technique support, resource support, and teacher support improve the frequency and effectiveness of interaction among group members, and a collaborative group with more members may increase the diversity of views, which is helpful to cultivate critical thinking utilizing the approach of collaborative problem-solving.

With regard to the learning scaffold, the three different kinds of learning scaffolds can all enhance critical thinking. Among them, the teacher-supported learning scaffold has the largest overall effect size, demonstrating the interdependence of effective learning scaffolds and collaborative problem-solving. This outcome is in line with some research findings; as an example, a successful strategy is to encourage learners to collaborate, come up with solutions, and develop critical thinking skills by using learning scaffolds (Reiser, 2004 ; Xu et al., 2022 ); learning scaffolds can lower task complexity and unpleasant feelings while also enticing students to engage in learning activities (Wood et al., 2006 ); learning scaffolds are designed to assist students in using learning approaches more successfully to adapt the collaborative problem-solving process, and the teacher-supported learning scaffolds have the greatest influence on critical thinking in this process because they are more targeted, informative, and timely (Xu et al., 2022 ).

With respect to the measuring tool, despite the fact that standardized measurement tools (such as the WGCTA, CCTT, and CCTST) have been acknowledged as trustworthy and effective by worldwide experts, only 54.43% of the research included in this meta-analysis adopted them for assessment, and the results indicated no intergroup differences. These results suggest that not all teaching circumstances are appropriate for measuring critical thinking using standardized measurement tools. “The measuring tools for measuring thinking ability have limits in assessing learners in educational situations and should be adapted appropriately to accurately assess the changes in learners’ critical thinking.”, according to Simpson and Courtney ( 2002 , p. 91). As a result, in order to more fully and precisely gauge how learners’ critical thinking has evolved, we must properly modify standardized measuring tools based on collaborative problem-solving learning contexts.

With regard to the subject area, the comprehensive effect size of science departments (e.g., mathematics, science, medical science) is larger than that of language arts and social sciences. Some recent international education reforms have noted that critical thinking is a basic part of scientific literacy. Students with scientific literacy can prove the rationality of their judgment according to accurate evidence and reasonable standards when they face challenges or poorly structured problems (Kyndt et al., 2013 ), which makes critical thinking crucial for developing scientific understanding and applying this understanding to practical problem solving for problems related to science, technology, and society (Yore et al., 2007 ).

Suggestions for critical thinking teaching

Other than those stated in the discussion above, the following suggestions are offered for critical thinking instruction utilizing the approach of collaborative problem-solving.

First, teachers should put a special emphasis on the two core elements, which are collaboration and problem-solving, to design real problems based on collaborative situations. This meta-analysis provides evidence to support the view that collaborative problem-solving has a strong synergistic effect on promoting students’ critical thinking. Asking questions about real situations and allowing learners to take part in critical discussions on real problems during class instruction are key ways to teach critical thinking rather than simply reading speculative articles without practice (Mulnix, 2012 ). Furthermore, the improvement of students’ critical thinking is realized through cognitive conflict with other learners in the problem situation (Yang et al., 2008 ). Consequently, it is essential for teachers to put a special emphasis on the two core elements, which are collaboration and problem-solving, and design real problems and encourage students to discuss, negotiate, and argue based on collaborative problem-solving situations.

Second, teachers should design and implement mixed courses to cultivate learners’ critical thinking, utilizing the approach of collaborative problem-solving. Critical thinking can be taught through curriculum instruction (Kuncel, 2011 ; Leng and Lu, 2020 ), with the goal of cultivating learners’ critical thinking for flexible transfer and application in real problem-solving situations. This meta-analysis shows that mixed course teaching has a highly substantial impact on the cultivation and promotion of learners’ critical thinking. Therefore, teachers should design and implement mixed course teaching with real collaborative problem-solving situations in combination with the knowledge content of specific disciplines in conventional teaching, teach methods and strategies of critical thinking based on poorly structured problems to help students master critical thinking, and provide practical activities in which students can interact with each other to develop knowledge construction and critical thinking utilizing the approach of collaborative problem-solving.

Third, teachers should be more trained in critical thinking, particularly preservice teachers, and they also should be conscious of the ways in which teachers’ support for learning scaffolds can promote critical thinking. The learning scaffold supported by teachers had the greatest impact on learners’ critical thinking, in addition to being more directive, targeted, and timely (Wood et al., 2006 ). Critical thinking can only be effectively taught when teachers recognize the significance of critical thinking for students’ growth and use the proper approaches while designing instructional activities (Forawi, 2016 ). Therefore, with the intention of enabling teachers to create learning scaffolds to cultivate learners’ critical thinking utilizing the approach of collaborative problem solving, it is essential to concentrate on the teacher-supported learning scaffolds and enhance the instruction for teaching critical thinking to teachers, especially preservice teachers.

Implications and limitations

There are certain limitations in this meta-analysis, but future research can correct them. First, the search languages were restricted to English and Chinese, so it is possible that pertinent studies that were written in other languages were overlooked, resulting in an inadequate number of articles for review. Second, these data provided by the included studies are partially missing, such as whether teachers were trained in the theory and practice of critical thinking, the average age and gender of learners, and the differences in critical thinking among learners of various ages and genders. Third, as is typical for review articles, more studies were released while this meta-analysis was being done; therefore, it had a time limit. With the development of relevant research, future studies focusing on these issues are highly relevant and needed.

Conclusions

The subject of the magnitude of collaborative problem-solving’s impact on fostering students’ critical thinking, which received scant attention from other studies, was successfully addressed by this study. The question of the effectiveness of collaborative problem-solving in promoting students’ critical thinking was addressed in this study, which addressed a topic that had gotten little attention in earlier research. The following conclusions can be made:

Regarding the results obtained, collaborative problem solving is an effective teaching approach to foster learners’ critical thinking, with a significant overall effect size (ES = 0.82, z  = 12.78, P  < 0.01, 95% CI [0.69, 0.95]). With respect to the dimensions of critical thinking, collaborative problem-solving can significantly and effectively improve students’ attitudinal tendency, and the comprehensive effect is significant (ES = 1.17, z  = 7.62, P  < 0.01, 95% CI [0.87, 1.47]); nevertheless, it falls short in terms of improving students’ cognitive skills, having only an upper-middle impact (ES = 0.70, z  = 11.55, P  < 0.01, 95% CI [0.58, 0.82]).

As demonstrated by both the results and the discussion, there are varying degrees of beneficial effects on students’ critical thinking from all seven moderating factors, which were found across 36 studies. In this context, the teaching type (chi 2  = 7.20, P  < 0.05), intervention duration (chi 2  = 12.18, P  < 0.01), subject area (chi 2  = 13.36, P  < 0.05), group size (chi 2  = 8.77, P  < 0.05), and learning scaffold (chi 2  = 9.03, P  < 0.01) all have a positive impact on critical thinking, and they can be viewed as important moderating factors that affect how critical thinking develops. Since the learning stage (chi 2  = 3.15, P  = 0.21 > 0.05) and measuring tools (chi 2  = 0.08, P  = 0.78 > 0.05) did not demonstrate any significant intergroup differences, we are unable to explain why these two factors are crucial in supporting the cultivation of critical thinking in the context of collaborative problem-solving.

Data availability

All data generated or analyzed during this study are included within the article and its supplementary information files, and the supplementary information files are available in the Dataverse repository: https://doi.org/10.7910/DVN/IPFJO6 .

Bensley DA, Spero RA (2014) Improving critical thinking skills and meta-cognitive monitoring through direct infusion. Think Skills Creat 12:55–68. https://doi.org/10.1016/j.tsc.2014.02.001

Article   Google Scholar  

Castle A (2009) Defining and assessing critical thinking skills for student radiographers. Radiography 15(1):70–76. https://doi.org/10.1016/j.radi.2007.10.007

Chen XD (2013) An empirical study on the influence of PBL teaching model on critical thinking ability of non-English majors. J PLA Foreign Lang College 36 (04):68–72

Google Scholar  

Cohen A (1992) Antecedents of organizational commitment across occupational groups: a meta-analysis. J Organ Behav. https://doi.org/10.1002/job.4030130602

Cooper H (2010) Research synthesis and meta-analysis: a step-by-step approach, 4th edn. Sage, London, England

Cindy HS (2004) Problem-based learning: what and how do students learn? Educ Psychol Rev 51(1):31–39

Duch BJ, Gron SD, Allen DE (2001) The power of problem-based learning: a practical “how to” for teaching undergraduate courses in any discipline. Stylus Educ Sci 2:190–198

Ennis RH (1989) Critical thinking and subject specificity: clarification and needed research. Educ Res 18(3):4–10. https://doi.org/10.3102/0013189x018003004

Facione PA (1990) Critical thinking: a statement of expert consensus for purposes of educational assessment and instruction. Research findings and recommendations. Eric document reproduction service. https://eric.ed.gov/?id=ed315423

Facione PA, Facione NC (1992) The California Critical Thinking Dispositions Inventory (CCTDI) and the CCTDI test manual. California Academic Press, Millbrae, CA

Forawi SA (2016) Standard-based science education and critical thinking. Think Skills Creat 20:52–62. https://doi.org/10.1016/j.tsc.2016.02.005

Halpern DF (2001) Assessing the effectiveness of critical thinking instruction. J Gen Educ 50(4):270–286. https://doi.org/10.2307/27797889

Hu WP, Liu J (2015) Cultivation of pupils’ thinking ability: a five-year follow-up study. Psychol Behav Res 13(05):648–654. https://doi.org/10.3969/j.issn.1672-0628.2015.05.010

Huber K (2016) Does college teach critical thinking? A meta-analysis. Rev Educ Res 86(2):431–468. https://doi.org/10.3102/0034654315605917

Kek MYCA, Huijser H (2011) The power of problem-based learning in developing critical thinking skills: preparing students for tomorrow’s digital futures in today’s classrooms. High Educ Res Dev 30(3):329–341. https://doi.org/10.1080/07294360.2010.501074

Kuncel NR (2011) Measurement and meaning of critical thinking (Research report for the NRC 21st Century Skills Workshop). National Research Council, Washington, DC

Kyndt E, Raes E, Lismont B, Timmers F, Cascallar E, Dochy F (2013) A meta-analysis of the effects of face-to-face cooperative learning. Do recent studies falsify or verify earlier findings? Educ Res Rev 10(2):133–149. https://doi.org/10.1016/j.edurev.2013.02.002

Leng J, Lu XX (2020) Is critical thinking really teachable?—A meta-analysis based on 79 experimental or quasi experimental studies. Open Educ Res 26(06):110–118. https://doi.org/10.13966/j.cnki.kfjyyj.2020.06.011

Liang YZ, Zhu K, Zhao CL (2017) An empirical study on the depth of interaction promoted by collaborative problem solving learning activities. J E-educ Res 38(10):87–92. https://doi.org/10.13811/j.cnki.eer.2017.10.014

Lipsey M, Wilson D (2001) Practical meta-analysis. International Educational and Professional, London, pp. 92–160

Liu Z, Wu W, Jiang Q (2020) A study on the influence of problem based learning on college students’ critical thinking-based on a meta-analysis of 31 studies. Explor High Educ 03:43–49

Morris SB (2008) Estimating effect sizes from pretest-posttest-control group designs. Organ Res Methods 11(2):364–386. https://doi.org/10.1177/1094428106291059

Article   ADS   Google Scholar  

Mulnix JW (2012) Thinking critically about critical thinking. Educ Philos Theory 44(5):464–479. https://doi.org/10.1111/j.1469-5812.2010.00673.x

Naber J, Wyatt TH (2014) The effect of reflective writing interventions on the critical thinking skills and dispositions of baccalaureate nursing students. Nurse Educ Today 34(1):67–72. https://doi.org/10.1016/j.nedt.2013.04.002

National Research Council (2012) Education for life and work: developing transferable knowledge and skills in the 21st century. The National Academies Press, Washington, DC

Niu L, Behar HLS, Garvan CW (2013) Do instructional interventions influence college students’ critical thinking skills? A meta-analysis. Educ Res Rev 9(12):114–128. https://doi.org/10.1016/j.edurev.2012.12.002

Peng ZM, Deng L (2017) Towards the core of education reform: cultivating critical thinking skills as the core of skills in the 21st century. Res Educ Dev 24:57–63. https://doi.org/10.14121/j.cnki.1008-3855.2017.24.011

Reiser BJ (2004) Scaffolding complex learning: the mechanisms of structuring and problematizing student work. J Learn Sci 13(3):273–304. https://doi.org/10.1207/s15327809jls1303_2

Ruggiero VR (2012) The art of thinking: a guide to critical and creative thought, 4th edn. Harper Collins College Publishers, New York

Schellens T, Valcke M (2006) Fostering knowledge construction in university students through asynchronous discussion groups. Comput Educ 46(4):349–370. https://doi.org/10.1016/j.compedu.2004.07.010

Sendag S, Odabasi HF (2009) Effects of an online problem based learning course on content knowledge acquisition and critical thinking skills. Comput Educ 53(1):132–141. https://doi.org/10.1016/j.compedu.2009.01.008

Sison R (2008) Investigating Pair Programming in a Software Engineering Course in an Asian Setting. 2008 15th Asia-Pacific Software Engineering Conference, pp. 325–331. https://doi.org/10.1109/APSEC.2008.61

Simpson E, Courtney M (2002) Critical thinking in nursing education: literature review. Mary Courtney 8(2):89–98

Stewart L, Tierney J, Burdett S (2006) Do systematic reviews based on individual patient data offer a means of circumventing biases associated with trial publications? Publication bias in meta-analysis. John Wiley and Sons Inc, New York, pp. 261–286

Tiwari A, Lai P, So M, Yuen K (2010) A comparison of the effects of problem-based learning and lecturing on the development of students’ critical thinking. Med Educ 40(6):547–554. https://doi.org/10.1111/j.1365-2929.2006.02481.x

Wood D, Bruner JS, Ross G (2006) The role of tutoring in problem solving. J Child Psychol Psychiatry 17(2):89–100. https://doi.org/10.1111/j.1469-7610.1976.tb00381.x

Wei T, Hong S (2022) The meaning and realization of teachable critical thinking. Educ Theory Practice 10:51–57

Xu EW, Wang W, Wang QX (2022) A meta-analysis of the effectiveness of programming teaching in promoting K-12 students’ computational thinking. Educ Inf Technol. https://doi.org/10.1007/s10639-022-11445-2

Yang YC, Newby T, Bill R (2008) Facilitating interactions through structured web-based bulletin boards: a quasi-experimental study on promoting learners’ critical thinking skills. Comput Educ 50(4):1572–1585. https://doi.org/10.1016/j.compedu.2007.04.006

Yore LD, Pimm D, Tuan HL (2007) The literacy component of mathematical and scientific literacy. Int J Sci Math Educ 5(4):559–589. https://doi.org/10.1007/s10763-007-9089-4

Zhang T, Zhang S, Gao QQ, Wang JH (2022) Research on the development of learners’ critical thinking in online peer review. Audio Visual Educ Res 6:53–60. https://doi.org/10.13811/j.cnki.eer.2022.06.08

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Acknowledgements

This research was supported by the graduate scientific research and innovation project of Xinjiang Uygur Autonomous Region named “Research on in-depth learning of high school information technology courses for the cultivation of computing thinking” (No. XJ2022G190) and the independent innovation fund project for doctoral students of the College of Educational Science of Xinjiang Normal University named “Research on project-based teaching of high school information technology courses from the perspective of discipline core literacy” (No. XJNUJKYA2003).

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Xu, E., Wang, W. & Wang, Q. The effectiveness of collaborative problem solving in promoting students’ critical thinking: A meta-analysis based on empirical literature. Humanit Soc Sci Commun 10 , 16 (2023). https://doi.org/10.1057/s41599-023-01508-1

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Top 10 Challenges to Teaching Math and Science Using Real Problems

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Nine in ten educators believe that using a problem-solving approach to teaching math and science can be motivating for students, according to an EdWeek Research Center survey.

But that doesn’t mean it’s easy.

Teachers perceive lack of time as a big hurdle. In fact, a third of educators—35 percent—worry that teaching math or science through real-world problems—rather than focusing on procedures—eats up too many precious instructional minutes.

Other challenges: About another third of educators said they weren’t given sufficient professional development in how to teach using a real-world problem-solving approach. Nearly a third say reading and writing take priority over STEM, leaving little bandwidth for this kind of instruction. About a quarter say that it’s tough to find instructional materials that embrace a problem-solving perspective.

Nearly one in five cited teachers’ lack of confidence in their own problem solving, the belief that this approach isn’t compatible with standardized tests, low parent support, and the belief that student behavior is so poor that this approach would not be feasible.

The nationally representative survey included 1,183 district leaders, school leaders, and teachers, and was conducted from March 27 to April 14. (Note: The chart below lists 11 challenges because the last two on the list—dealing with teacher preparation and student behavior—received the exact percentage of responses.)

Trying to incorporate a problem-solving approach to tackling math can require rethinking long-held beliefs about how students learn, said Elham Kazemi, a professor in the teacher education program at the University of Washington.

Most teachers were taught math using a procedural perspective when they were in school. While Kazemi believes that approach has merit, she advocates for exposing students to both types of instruction.

Many educators have “grown up around a particular model of thinking of teaching and learning as the teacher in the front of the room, imparting knowledge, showing kids how to do things,” Kazemi said.

To be sure, some teachers have figured out how to incorporate some real-world problem solving alongside more traditional methods. But it can be tough for their colleagues to learn from them because “teachers don’t have a lot of time to collaborate with one another and see each other teach,” Kazemi said.

What’s more, there are limited instructional materials emphasizing problem solving, Kazemi said.

Though that’s changing, many of the resources available have “reinforced the idea that the teacher demonstrates solutions for kids,” Kazemi said.

Molly Daley, a regional math coordinator for Education Service District 112, which serves about 30 districts near Vancouver, Wash., has heard teachers raise concerns that teaching math from a problem-solving perspective takes too long—particularly given the pressure to get through all the material students will need to perform well on state tests.

Daley believes, however, that being taught to think about math in a deeper way will help students tackle math questions on state assessments that may look different from what they’ve seen before.

“It’s myth that it’s possible to cover everything that will be on the test,” as it will appear, she said. “There’s actually no way to make sure that kids have seen every single possible thing the way it will show up. That’s kind of a losing proposition.”

But rushing through the material in a purely procedural way may actually be counterproductive, she said.

Teachers don’t want kids to “sit down at the test and say, ‘I haven’t seen this and therefore I can’t do it,’” Daley said. “I think a lot of times teachers can unintentionally foster that because they’re so urgently trying to cover everything. That’s where the kind of mindless [teaching] approaches come in.”

Teachers may think to themselves: “’OK, I’m gonna make this as simple as possible, make sure everyone knows how to follow the steps and then when they see it, they can follow it,” Daley said.

But that strategy might “take away their students’ confidence that they can figure out what to do when they don’t know what to do, which is really what you want them to be thinking when they go to approach a test,” Daley said.

Data analysis for this article was provided by the EdWeek Research Center. Learn more about the center’s work.

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Problem solving in mathematics education: tracing its foundations and current research-practice trends

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In tracing recent research trends and directions in mathematical problem-solving, it is argued that advances in mathematics practices occur and take place around two intertwined activities, mathematics problem formulation and ways to approach and solve those problems. In this context, a problematizing principle emerges as central activity to organize mathematics curriculum proposals and ways to structure problem-solving learning environments. Subjects’ use of concrete, abstract, symbolic, or digital tools not only influences the ways to pose and pursue mathematical problems; but also shapes the type of representation, exploration, and reasoning they engage to work and solve problems. Problem-solving foundations that privilege learners’ development of habits of mathematical practices that involve an inquiry method to formulate conjectures, to look for different ways to represent and approach problems, and to support and communicate results shed light on directions of current research trends and the relevance of rethinking curriculum proposals and extending problem-solving environments in terms of teachers/students’ consistent use of digital tools and online developments.

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1 Introduction and rationale

Mathematical problem solving has been a prominent theme and research area in the mathematics education agenda during the last four decades. Problem-solving perspectives have influenced and shaped mathematics curriculum proposals and ways to support learning environments worldwide (Törner et al., 2007 ; Toh et al., 2023 ). Various disciplinary communities have identified and contributed to connect problem-solving approaches with the students’ learning, construction, and application of mathematical knowledge. The mathematics community recognizes that the formulation and resolution of problems are central activities in the development of the discipline (Halmos, 1980 , Polya, 1945 ). Indeed, the identification and presentation of lists of unsolved mathematical problems have been a tradition that has inspired the mathematics community to approach mathematical problems and to generate mathematical knowledge (Hilbert, 1902 ; Devlin, 2002 ). Thus, mathematical problems, results, and solution attempts provide information regarding what areas and contents were studied at different times during the development of the discipline (Santos-Trigo, 2020a , b ). Cai et al. ( 2023 ) stated that “ …[E]ngaging learners in the activity of problem posing reflects a potentially strong link to the discipline of mathematics” (p. 5). Thurston ( 1994 ) recognized that understanding and applying a mathematical concept implies analysing, coordinating, and integrating diverse meanings (geometric, visual, intuitive, and formal definition) associated with such concept and ways to carry out corresponding procedures and operations in problematic situations.

The centrality of problem-solving in mathematicians’ own work and in their teaching, is incontrovertible. Problem-solving is also a central topic for mathematics educators, who have developed conceptual frameworks to formulate general ideas about problem-solving (as opposed to the specific ideas needed for solving specific problems) (Fried, 2014 ; p.17).

That is, the mathematics education community is interested in analysing and documenting the students’ cognitive and social behaviours to understand and develop mathematical knowledge and problem-solving competencies. “…the idea of understanding how mathematicians treat and solve problems, and then implementing this understanding in instruction design, was pivotal in mathematics education research and practice” (Koichu, 2014 ). In addition, other disciplines such as psychology, cognitive science or artificial intelligence have provided tools and methods to delve into learners’ ways to understand mathematical concepts and to work on problem situations. Thus, members of various communities have often worked in collaboration to identify and relate relevant aspects of mathematical practices with the design and implementation of learning scenarios that foster and enhance students’ mathematical thinking and the development of problem-solving competencies.

2 Methods and procedures

Research focus, themes, and inquiry methods in the mathematical problem-solving agenda have varied and been influenced and shaped by theoretical and methodological developments of mathematics education as a discipline (English & Kirshner, 2016 ; Liljedahl & Cai, 2021 ). Further, research designs and methods used in cognitive, social, and computational fields have influenced the ways in which mathematical problem-solving research are framed. An overarching question to capture shifts and foundations in problem-solving developments was: How has mathematical problem-solving research agenda varied and evolved in terms of ways to frame, pose, and pursue research questions? In addressing this question, it was important to identify and contrast the structure and organization around some published problem-solving reviews (Lester, 1994 ; Törner et al., 2007 ; Rott et al., 2021 ; Liljedahl & Cai, 2021 ; Toh et al., 2023 ) to shed light on a possible route to connect seminal developments in the field with current research trends and perspectives in mathematical problem-solving developments. The goal was to identify common problem-solving principles that have provided a rational and foundations to support recent problem-solving approaches for learners to construct mathematical knowledge and to develop problem-solving competencies. The criteria to select the set of published peer-reviewed studies, to consider in this review, involved choosing articles published in indexed journals (ZDM-Mathematics Education, Educational Studies in Mathematics, Mathematical Thinking and Learning, Journal of Mathematical Behavior, and Journal for Research in Mathematics Education); contributions that appear in International Handbooks in Mathematics Education; and chapters published in recent mathematical problem-solving books. The initial search included 205 publications whose number was reduced to 55, all published in English, based on reviewing their abstracts and conclusions. Around 100 of the initial selection appeared in the references of an ongoing weekly mathematical problem-solving doctoral seminar that has been implemented during the last six years in our department. In addition, some well-known authors in the field were asked to identify their most representative publications to include in the review list. Here, some suggestions were received, but at the end the list of contributions, that appears in the references section, was chosen based on my vision and experience in the field. The goal was to identify main issues or dimensions to frame and analyse recent research trends and perspectives in mathematical problem-solving developments. Thus, seminal reviews in the field (Schoenfeld, 1992 ; Lester, 1994 ; Törner et al., 2007 ) provided directions on ways to structure and select the questions used to analyse the selected contributions. Table  1 shows chosen issues that resemble features of an adjusted framework that Lester ( 1994 ) proposed to organize, summarize, and analyse problem-solving developments in terms of research emphasis (themes and research questions), methodologies (research designs and methods), and achieved results that the problem-solving community addressed during the 1970–1994 period. Furthermore, relevant shifts in the mathematical problem-solving agenda could be identified and explained in terms of what the global mathematics education and other disciplines pursue at different periods.

It is important to mention that the content and structure of this paper involve a narrative synthesis of selected articles that includes contributions related to mathematical problem-solving foundations and those that address recent developments published in the last 9 years that involve the use of digital technologies. Table  1 shows themes, issues, and overarching questions that were used to delve into problem-solving developments.

To contextualize the current state of art in the field, it is important to revisit problem-solving principles and tenets that provide foundations and a rationale to centre and support the design and implementation of learning environments around problem-solving activities (Santos-Trigo, 2020a , b ). The identification of mathematical problem-solving foundations also implies acknowledging what terms, concepts, and language or discourse that the problem-solving community has used to refer to and frame problem-solving approaches. For example, routine and nonroutine tasks, heuristic and metacognitive strategies, students’ beliefs, mathematical thinking and practices, resources, orientations, etc. are common terms used to explain, foster, and characterize students’ problem-solving behaviours and performances. Recently, the consistent use of digital technologies in educational tasks has extended the problem-solving language to include terms such as subjects’ tool appropriation, dynamic models, dragging or moving orderly objects, tracing loci, visual or empirical solution, ChatGPT prompts, etc.

3 On mathematical problem-solving foundations and the problematizing principle

There might be different ways to interpret and implement a problem-solving approach for students to understand concepts and to solve problems (Törner, Schoenfeld, & Reiss, 2007 ; Toh et al., 2023 ); nevertheless, there are common principles or tenets that distinguish and support a problem-solving teaching/learning environment. A salient feature in any problem-solving approach to learn mathematics is a conceptualization of the discipline that privileges and enhance the students’ development of mathematical practices or reasoning habits of mathematical thinking (Cuoco, et, al., 1996 ; Dick & Hollebrands, 2011 ; Schoenfeld, 2022 ). In this context, students need to conceptualize and think of their own learning as a set of dilemmas that are represented, explored, and solved in terms of mathematical resources and strategies (Santos-Trigo, 2023 ; Hiebert et al., 1996 ).

Furthermore, students’ problem-solving experiences and behaviours reflect and become a way of thinking that is consistent with mathematics practices and is manifested in terms of the activities they engage throughout all problem-solving phases. Thus, they privilege the development of mathematics habits such as to always look for different ways to model and explore mathematical problems, to formulate conjectures, and to search for arguments to support them, share problem solutions, defend their ideas, and to develop a proper language to communicate results. In terms of connecting ways of developing mathematical knowledge and the design of learning environments to develop mathematical thinking and problem-solving competencies, Polya ( 1945 ) identifies an inquiry approach for students to understand, make sense, and apply mathematical concepts. He illustrated the importance for students to pose and pursue different questions around four intertwined problem-solving phases: Understanding and making sense of the problem statement (what is the problem about? What data are provided? What is asked to find? etc.), the design of a solution plan (how the problem can be approached? ), the implementation of such plan (how the plan can be achieved? ), and the looking-back phase that involves reviewing the solution process (data used, checking the involved operations, consistency of units, and partial and global solution), generalizing the solution methods and posing new problems. Indeed, the looking-back phase involves the formulation of new or related problems (Toh et al., 2023 ). “For Pólya, mathematics was about inquiry; it was about sense making; it was about understanding how and why mathematical ideas fit together the ways they do” (cited in Schoenfeld, 2020 , p. 1167).

Likewise, the Nobel laureate I. I. Rabi mentioned that, when he came home from school, “while other mothers asked their kids ‘ Did you learn anything today ?’ [my mother] would say, ‘ Izzy, did you ask a good question today ?’” (Berger, 2014 , p.67).

Thus, the problematizing principle is key for students to engage in mathematical problem-solving activities, and it gets activated by an inquiry or inquisitive method that is expressed in terms of questions that students pose and pursue to delve into concepts meaning, representations, explorations, operations, and to work on mathematical tasks (Santos-Trigo, 2020a , b ).

4 The importance of mathematical tasks and the role of tools in problem-solving perspectives

In a problem-solving approach, learners develop a way of thinking to work on different types of tasks that involve a variety of context and aims (Cai & Hwang, 2023 ). A task might require students to formulate a problem from given information, to estimate how much water a family spend in one year, to prove a geometry theorem, to model genetic sequences or to understand the interplay between climate and geography. In this process, students identify mathematical resources, concepts, and strategies to model and explore partial and global solutions, and ways to extend solution methods and results. Furthermore, mathematical tasks or problems are essential for students to engage in mathematical practice and to develop problem-solving competencies. Task statements should be situated in different contexts including realistic, authentic, or mathematical domains, and prompts or questions to solve or respond or even provide information or data for students to formulate and solve their own problems (problem posing). Current events or problematic situations such as climate change, immigration, or pandemics not only are part of individuals concerns; but also, a challenge for teachers and students to model and analyze those complex problems through mathematics and others disciplines knowledge (English, 2023 ). Santos-Trigo ( 2019 ) proposed a framework to transform exercises or routine textbook problems into a series of nonroutine tasks in which students have an opportunity to dynamically model, explore, and extend, the initial problem. Here, the use of technology becomes important to explore the behavior of some elements within the model to find objects’ mathematical relationships. That is, students work on tasks in such a way that even routine problems become a starting point for them to engage in mathematical reflection to extend the initial nature of the task (Santos-Trigo & Reyes-Martínez, 2019 ). Recently, the emergence of tools such as the ChatGPT has confirmed the importance for learners to problematize situations, including complex problems, in terms of providing prompts or inputs that the tool processes and answers. Here, students analyze the tool’ responses and assess its pertinence to work and solve the task. Indeed, a way to use ChatGPT involves that students understand or make sense of the problem statement and pose questions (inputs or prompts) to ask the tool for concept information or ways to approach or solve the task. Then, students analyze the relevance, viability, and consistency of the tool’s answer and introduce new inputs to continue with the solution process or to look for another way to approach the task. Based on the ChatGPT output or task solution, students could always ask whether the tool can provide other ways to solve the task.

5 Main problem-solving research themes and results

In this section the focus will be on identifying certain problem-solving developments that have permeated recent directions of the field. One relates to the importance of extending research designs to analyse and characterize learners’ problem-solving process to work on different types of tasks. Another development involves ways in which theoretical advances in mathematics education have shaped the mathematical problem-solving research agenda and the extent to which regional or national educational systems or traditions influence the developments of conceptual frameworks in the field and ways to implement problem-solving activities within the corresponding system. Finally, research results in the field have provided directions to design and implement curriculum proposals around the world and these proposals have evolved in terms of both content structure and classroom dynamics including the use of digital technologies. Santos-Trigo ( 2023 ) stated that the teachers and students’ systematic use of digital technologies not only expands their ways of reasoning and solving mathematical problems; but also opens new research areas that aim to analyse the integration of several digital tools in curriculum proposals and learning scenarios. The focus of this review will be on presenting problem-solving directions and results in the last 9 years; however, it became relevant to identify and review what principles and tenets provided bases or foundations to support and define current research trends and directions in the field. That is, accumulated research that has contributed to advance and expand the problem-solving research agenda included shifts in the tools used to delve into learners’ problem approaches, the development of conceptual frameworks to explain and characterize students’ mathematical thinking, the tools used to work on mathematical tasks (from paper and pencil, ruler and compass or semiotic tools to digital apps), and in the design of curriculum proposals and the implementation of problem-solving learning scenarios.

5.1 Relevant shifts in problem-solving developments and results

Questions used to analyse important developments in the field include: What research designs and tools are used to foster and analyse learners’ problem-solving performances? How have conceptual frameworks evolved to pose and frame research questions in the field? How have accumulated research results in the field been used to support curriculum proposals and their implementation?

5.1.1 Methodological and research paradigms

Research designs in problem-solving studies have gradually moved from quantitative or statistical paradigms to qualitative perspectives that involve data collection from different sources such as task-based interviews, fieldnotes from observations, students’ written reports, etc. to analyse students’ problem-solving approaches and performances. Trustworthiness of results included triangulating and interpreting data sources from students’ videotapes transcriptions, outside observer notes, class observations, etc. (Stake, 2000 ). Hence, the work of Krutestkii ( 1976 ) was seminal in providing tools to delve into the students’ thinking while solving mathematical tasks. His research program aimed to study the nature and structure of children’ mathematical abilities. His methodological approach involved the use of student’s task-based interviews, teachers, and mathematicians’ questionaries to explore the nature of mathematical abilities, the analysis of eminent mathematicians and physicists regarding their nature and emergence of their talents and case studies of gifted children in mathematics. A major contribution of his research was the variety of mathematical tasks used to explore and analyse the mathematical abilities of school children. Recently, the mathematical problem-posing agenda has been revisited to advance conceptual frameworks to enhance the students’ formulation of problems to learn concepts and to develop problem-solving competencies (Cai et al., 2023 ). In general, the initial qualitative research tendency privileged case studies where individual students were asked to work on mathematical tasks to document their problem-solving performances. Later, research designs include the students’ participation in small groups and the analysis of students’ collaboration with the entire group (Brady et al., 2023 ). Bricolage frameworks that share tenets and information from different fields have become a powerful tool for researchers to understand complex people’ problem-solving proficiency (Lester, 2005 ; English, 2023 ).

5.1.2 Theoretical developments in mathematics education

In mathematics education, the constructivism perspective became relevant to orient and support research programs. Specifically, the recognition that students construct mathematical concepts and ideas through active participation as a part of a learning community that fosters and values what they bring into the classroom (eliciting students’ understanding) and sharing and discussing with peers their ways to work on mathematical activities. Further, it was recognized that students’ learning of mathematics takes place within a sociocultural environment (situated learning) that promotes the students’ interaction in small groups, pairs, and whole group discussions. Thus, problem-solving environments transited from teachers being a main figure to organize learning activities and to model problem-solving behaviours to being centred on students’ active participation to work on a variety of mathematical tasks as a part of a learning community (Lester & Cai, 2016 ). English ( 2023 ) proposed A STEM-based problem-solving framework that addresses the importance of a multidisciplinary approach and experiences to work on complex problems. Here, students develop a system of inquiry that integrates critical thinking, mathematical modelling, and a creative and innovative approach to deal with problematic situations situated in contexts beyond school problems. The STEM-based problem-solving framework enhances and favours the students’ development of multidisciplinary thinking to formulate and approach challenging problematic situations. To this end, they need to problematize information to characterize local and global problems and to collaboratively work on feasible approaches and solutions. It integrates 21st century skills that include an inquiry problem-solving approach to develop and exhibit critical thinking, creativity, and innovative solutions.

5.1.3 Countries or regional education traditions and their influence on the problem-solving agenda

The emergence of problem-solving frameworks takes place within an educational and socio-cultural context that provides conditions for their development and dissemination, but also limitations in their applications inside the mathematics education community. Brady et al. ( 2023 ) pointed out that:

…shifts in the theoretical frameworks of mathematics education researchers favored a widening of the view on problem solving from information-processing theories toward sociocultural theories that encouraged a conception of problem-solving as situated cognition unfolding within a community of practice (p. 34).

In addition, regional or national educational systems and research traditions also shape the problem-solving research and practice agenda. For example, in France, problem-solving approaches and research are framed in terms of two relevant theoretical and practical frameworks: Theory of Didactic Situation and the Anthropological Theory of Didactics (Artigue & Houdement, 2007 ). While, in the Netherlands, problem-solving approaches are situated within the theory of Realistic Mathematics that encourages and supports the students’ construction of meaning of concepts and methods in terms of modelling real-life and mathematical situations (Doorman et al., 2007 ). Ding et al. ( 2022 ) stated that the Chinese educational system refers to problem solving as an instructional goal and an approach to learn mathematics. Here, students deal with different types of problem-solving activities that include finding multiple solutions to one problem, one solution to multiple problems, and one problem multiple changes. Thus, ‘teaching with variation’ is emphasized in Chinese instruction in terms of “variations in solutions, presentations, and conditions/conclusions” (p. 482). Cai and Rott ( 2023 ) proposed a general problem-posing process model that distinguishes four problem-posing phases: Orientation (understanding the situation and what is required or is asked to pose); Connection that involves finding out or generating ideas and strategies to pose problems in different ways such as varying the given situation, or posing new problems; Generation refers to making the posed problem visible for others to understand it; and Reflection involves reflecting on her/his own process to pose the problem including ways to improve problem statements. The challenge in this model is to make explicit how the use of digital technologies can contribute to providing conditions for students to engage in all phases around problem- posing process.

5.1.4 Curriculum proposals and problem-solving teaching/learning scenarios

In the USA, the Common Core State Mathematics Standards curriculum proposal (CCSMS) identifies problem solving as a process standard that supports core mathematical practices that involve reasoning and proof, communication, representation, and connections. Thus, making sense of problems and persevering in solving them, reasoning abstractly and quantitatively, constructing viable arguments and critiquing the reasoning of others, modelling with mathematics, etc. are essential activities for students to develop mathematics proficiency and problem-solving approaches (Schoenfeld, 2023 ). In Singapore, the curriculum proposal identifies problem solving as the centre of its curriculum framework that relates its development with the study of concepts, skills, processes, attitudes, and metacognition (Lee et al., 2019 ). Recently, educational systems have begun to reform curriculum proposals to relate what the use of digital technologies demands in terms of selecting and structuring mathematical contents and ways to extend instructional settings (Engelbrecht & Borba, 2023 ). Indeed, Engelbrecht et al. ( 2023 ) identify what they call a classroom in movement or a distributed classroom - that transforms traditional cubic spaces to study the discipline into a movable setting that might combine remote and face-to-face students work.

It is argued that previous results in mathematical problem-solving research not only have contributed to recognize what is relevant and what common tenets distinguish and support problem-solving approaches; but also have provided bases to identify and pursue current problem-solving developments and directions. Hence, the consistent and coordinated use of several digital technologies and online developments (teaching and learning platforms) has opened new routes for learners to represent, explore, and work on mathematical problems; and to engage them in mathematical discussions beyond formal class settings. How does the students’ use of digital technologies expand the ways they reason and solve mathematical problems? What changes in classroom environments and physical settings are needed to recognize and include students’ face-to-face and remote work? (Engelbrecht et al., 2023 ).

In the next sections, the goal is to characterize the extent to which the consistent use of digital technologies and online developments provides affordances to restructure mathematical curriculum proposals and classrooms or learning settings and to enhance and expand students’ mathematical reasoning.

6 Current mathematical problem-solving trends and developments: the use of digital technologies

Although the use of technologies has been a recurrent theme in research studies, curriculum proposals, and teaching practices in mathematics education; during the COVID-pandemic lockdown, all teachers and students relied on digital technologies to work on mathematical tasks. At different phases, they developed and implemented not only novel paths to present, discuss, and approach teaching/learning activities; but also, ways to monitor and assess students’ problem-solving performances. When schools returned to teachers and students’ face-to-face activities, some questions emerged: What adjustments or changes in school practices are needed to consider and integrate those learning experiences that students developed during the social confinement? What digital tools should teachers and students use to work on mathematical tasks? How should teaching/learning practices reconcile students remote and face-to-face work? To address these questions, recent studies that involve ways to integrate technology in educational practices were reviewed, and their main themes and findings are organized and problematized to shed light on what the use of digital technologies contributes to frame and support learning environments.

6.1 The use of technology to reconceptualize students mathematical learning

There are different studies that document the importance and ways in which the students’ use of tools such as CAS or Excel offers an opportunity for them to think of concepts and problems in terms of different representations to transit from intuitive, visual, or graphic to formal or analytical reasoning (Arcavi et al., 2017 ). Others digital technologies, such as a Dynamic Geometry System Footnote 1 DGS, provide affordances for students to dynamically represent and explore mathematical problems. In students’ use of digital technologies, the problematizing principle becomes relevant to transform the tool into an instrument to work on mathematical tasks. Santos-Trigo ( 2019 ) provides examples where students rely on GeoGebra affordances to reconstruct figures that are given in problem statements; to transform routine problem into an investigation task; to model and explore tasks that involve variational reasoning; and to construct dynamic configurations to formulate and support mathematical relations. In this process, students not only exhibit diverse problem-solving strategies; but also, identify and integrate and use different concepts and resources that are studied in algebra, geometry, and calculus. That is, the use of technology provides an opportunity for students to integrate and connect knowledge from diverse areas or domains. For instance, Sinclair and Ferrara ( 2023 ) used the multi-touch application (TouchCounts) for children to work on mathematical challenging tasks.

6.2 The use of digital technologies to design a didactic route

There is indication, that the use of digital technologies offers different paths for students to learn mathematics (Leung & Bolite-Frant, 2015 ; Leung & Baccaglini-Frank, 2017 ). For instance, in the construction of a dynamic model of a problem, they are required to think of concepts and information embedded in the problem in terms of geometric representation or meaning. Thus, focusing on ways for students to represent and explore concepts geometrically could be the departure point to understand concepts and to solve mathematical problems. In addition, students can explore problems’ dynamic models (dragging schemes) in terms of visual, empirical, and graphic representations to initially identify relations that become relevant to approach and solve the problems. Thus, tool affordances become relevant for students to detect patterns, to formulate conjectures and to transit from empirical to formal argumentation to support problem solutions (Pittalis & Drijvers, 2023 ). Engelbrecht and Borba ( 2023 ) recognized that the prominent use of digital technologies in school mathematics has produced pedagogical shifts in teaching and learning practices to “encourage more active students learning, foster greater engagement, and provide more flexible access to learning’ (p. 1). Multiple use technologies such as internet, communication apps (ZOOM, Teams, Google Meet, etc.) become essential tools for teachers and students to present, communicate, and share information or to collaborate with peers. While tools used to represent, explore, and delve into concepts and to work and solve mathematical problems (Dynamic Geometry Systems, Wolframalpha, etc.) expand the students’ ways of reasoning and solving problems. Both types of technologies are not only important for teachers and students to continue working on school tasks beyond formal settings, but they also provide students with an opportunity to consult online resources such as Wikipedia or KhanAcademy to review or extend their concepts understanding, to analyse solved problems, and to contrast their teachers’ explanation of themes or concepts with those provided in learning platforms.

6.3 Students’ access to mathematics learning

Nowadays, cell phones are essential tools for people or students to interact or to approach diverse tasks and an educational challenge is how teachers/students can use them to work on mathematical tasks. During the COVID-19 social confinement, students relied on communication apps not only to interact with their teachers during class lectures; but also, to keep discussing tasks with peers beyond formal class meetings. That is, students realized that with the use of technology they could expand their learning space to include sharing and discussing ideas and problem solutions with peers beyond class sessions, consulting online learning platforms or material to review or extend their concepts understanding, and to watch videos to contrast experts’ concepts explanations and those provided by their teachers. In this perspective, the use of digital technologies increases the students’ access to different resources and the ways to work on mathematical tasks. Thus, available digital developments seem to extend the students collaborative work in addition to class activities. Furthermore, the flipped classroom model seems to offer certain advantages for students to learn the discipline and this model needs to be analysed in terms of what curriculum changes and ways to assess or monitor students learning are needed in its design and implementation (Cevikbas & Kaiser, 2022 ).

6.4 Changes in curriculum and mathematical assessment

It is recognized that the continuous development and availability of digital technologies is not only altering the ways in which individuals interact and face daily activities; but is also transforming educational practices and settings. Likewise, people’s concerns about multiple events or global problems such climate change, immigration, educational access, renewable resources, or racial conflicts or wars are themes that permeate the educational arena. Thus, curriculum reforms should address ways to connect students’ education with the analysis of these complex problems. English ( 2023 ) stated that:

The ill-defined problems of today, coupled with unexpected disruptions across all walks of life, demand advanced problem-solving by all citizens. The need to update outmoded forms of problem solving, which fail to take into account increasing global challenges, has never been greater (p.5).

In this perspective, mathematics curriculum needs to be structured around essential contents and habits of mathematical thinking for students to understand and make sense of real-world events that lead them to formulate, represent, and deal with a variety of problem situations. “Educators now increasingly seek to emphasise the practical applications of mathematics, such as modelling real-life scenarios and understanding statistical data (Engelbrecht & Borba, 2023 , p. 7). For instance, during the pandemic it was important to problematize the available data to follow, analyze and predict its spread behavior and to propose health measures to reduce people contagion. Thus, exponential functions, graphics, and their interpretations, data analysis, etc. were important mathematics content to understand the pandemic phenomena. Drijvers and Sinclair ( 2023 ) recognized that features of computational thinking share common grounds with mathematical thinking in terms of problem-solving activities that privilege model construction, the use of algorithms, abstraction processes and generalization of results. Thus, “a further integration of computational thinking in the mathematics curriculum is desirable”. In terms of ways to assess and monitor students’ learning, the idea is that with the use of a digital tool (digital wall or log), students could organize, structure, register, and monitor their individual and group work and learning experiences. That is, they could periodically report and share what difficulties they face to understand concepts or to work on a task, what questions they posed, what sources consult, etc. The information that appears in the digital wall is shared within the group and the teacher and students can provide feedback or propose new ideas or solutions (Santos-Trigo et al., 2022 ).

6.5 The integration of technologies and the emergence of conceptual frameworks

Institutions worldwide, in general, are integrating the use of different technologies in their educational practices, and they face the challenge to reconcile previous pandemic models and post confinement learning scenarios. “A pedagogical reason for using technology is to empower learners with extended or amplified abilities to acquire knowledge…technology can empower their cognitive abilities to reason in novice ways (Leung, 2011 , p. 327). Drijvers and Sinclair ( 2023 ) proposed a five-dimensional framework to delve into the rationale and purposes for the mathematics education community to integrate the use of digital technologies in mathematical teaching environments and students learning. The five interrelated categories address issues regarding how teachers and students’ use of digital technology contributes to reconceptualize and improve mathematics learning; to understand and explain how students’ mathematics learning develops; to design environments for mathematics learning; to foster and provide equitable access to mathematics learning; and to change mathematics curricula and teaching and assessment practices (Drijvers & Sinclair, 2023 ). Schoenfeld ( 2022 ) stated that “The challenge is to create robust learning environments that support every student in developing not only the knowledge and practices that underlie effective mathematical thinking, but that help them develop the sense of agency to engage in sense making” (p. 764). Højsted et al. ( 2022 ) argue about the importance of adjusting theoretical frameworks to explicitly integrate the use of digital technologies such as DGS and Computer Algebra Systems (CAS) in teaching practices. They referred to the Danish “Competencies and Mathematical Learning framework” (KOM) that gets articulated through tenets associated with the Theory of Instrumental Orchestration (TIO) and the notion of Justification Mediation (JM). In general terms, the idea is that learners get explicitly involved in a tool’ appropriation process that transforms the artifact into an instrument to understand concepts and to solve mathematical problems. That is, learners’ tool appropriation involves the development of cognitive schemata to rely on technology affordances to work on mathematical tasks. Koichu et al. ( 2022 ) pointed out that the incorporation of problem-solving approaches in instruction should be seen as a specific case of implementing innovation. To this end, they proposed a framework of problem-solving implementation chain that involves “a sequence of actions and interactions beginning with the development of a PS resource by researchers, which teachers then engage with in professional development (PD), and finally, teachers and students make use of in classrooms” (p. 4). In this case, problem-solving resources include the design of problematic situations (tasks) to engage students in mathematical discussions to make sense of problem statements or to ask them to pose a task.

7 Reflections and concluding remarks

Throughout different periods, the research and practice mathematical problem-solving agenda has contributed significantly to understand not only essentials in mathematical practices; but also, the development of conceptual frameworks to explain and document subjects’ cognitive, social, and affective behaviours to understand mathematical concepts and to develop problem-solving competencies. Leikin and Guberman ( 2023 ) pointed out that “…problem-solving is an effective didactical tool that allows pupils to mobilize their existing knowledge, construct new mathematical connections between known concepts and properties, and construct new knowledge in the process of overcoming challenges embedded in the problems” (p. 325). The study of people cognitive functioning to develop multidisciplinary knowledge and to solve problems involves documenting ways in which individuals make decisions regarding ways to organize their subject or disciplinary learning (how to interact with teachers or experts and peers; what material to consult, what tools to use, how to monitor their own learning, etc.) and to engage in disciplinary practices to achieve their learning goals. Both strategic and tactic decisions shape teachers and students’ ways to work on mathematical tasks. Kahneman ( 2011 ) shed light on how human beings make decisions to deal with questions and problematic situations. He argues that individuals rely on two systems to make decisions and engage in thinking processes; system one (fast thinking) that involves automatic, emotional, instinctive reasoning and system two (slow thinking) that includes logical, deliberative, effortful, or conscious reasoning. In educational tasks, the idea is that teachers and students develop experiences based on the construction and activation of system two. Thus, how teachers/students decide what tools or digital developments to use to work on mathematical problems becomes a relevant issue to address in the mathematics education agenda. Recent and consistent developments and the availability of digital technologies open novel paths for teachers and students to represent, explore, and approach mathematical tasks and, provide different tools to extend students and teachers’ mathematical discussions beyond classroom settings. In this perspective, it becomes important to discuss what changes the systematic use of digital technologies bring to the mathematics contents and to the ways to frame mathematical instruction. For example, the use of a Dynamic Geometry System to model and explore calculus, geometry or algebra classic problems dynamically not only offer students an opportunity to connect foundational concepts such as rate of change or the perpendicular bisector concept to geometrically study variational phenomena or conic sections; but also, to engage them in problem-posing activities (Santos-Trigo et al., 2021 ). Thus, teachers need to experience themselves different ways to use digital technologies to work on mathematical tasks and to identify instructional paths for students to internalize the use of digital apps as an instrument to understand concepts and to pose and formulate mathematical problems. Specifically, curriculum proposal should be structured around the development of foundational concepts and problem-solving strategies to formulate and pursue complex problems such as those involving climate changes, wealth distribution, immigration, pollution, mobility, connectivity, etc. To formulate and approach these problems, students need to develop a multidisciplinary thinking and rely on different tools to represent, explore, and share and continuously report partial solutions. To this end, they are encouraged to work with peers and groups as a part of learning community that fosters and values collective problem solutions. Finding multiple paths to solve problems becomes important for students to develop creative and innovative problem solutions (Leikin & Guberman, 2023 ). In this perspective, learning environments should provide conditions for students to transform digital applications in problem-solving tools to work on problematic situations. Online students’ assignments become an important component to structure and organize students and teachers’ face-to-face interactions. Likewise, the use of technology can also provide a tool for students to register and monitor their work and learning experiences. A digital wall or a problem-solving digital notebook (Santos-Trigo et al., 2022 ) could be introduced for students to register and monitor their learning experiences. Here, Students are asked to record on a weekly basis their work, questions, comments, and ideas that include: Questions they pose to understand concepts and problem statements; online resources and platforms they consult to contextualize problems and review and extend their understanding of involved concepts; concepts and strategies used to solve problems through different approaches; the Identification of other problems that can be solved with the methods that were used to solve the problem; digital technologies and online resources used to work on and solve the problem; dynamic models used to solve the problem and strategies used to identify and explore mathematical relations (dragging objects, measuring object attributes, tracing loci, using sliders, etc.; the formulation of new related problems including possible extensions for the initial problem; discussion of solutions of some new problems; and short recorded video presentation of their work and problem solutions. That is, the digital wall becomes an space for learners to share their work and to contrast and reflect on their peers work including extending their problem-solving approaches based on their teachers feedback and peers’ ideas or solutions.

The term Dynamic Geometry System is used, instead of Dynamic Geometry Environment or Dynamic Geometry Software, to emphasize that the app or tool interface encompasses a system of affordances that combines the construction of dynamic models, the use of Computer Algebra Systems and the use spreadsheet programs.

Arcavi, A., Drijvers, P., & Stacy, K. (2017). The learning and teaching of algebra. Ideas, insights, and activities . NY: Routledge. ISBN 9780415743723.

Artigue, M., & Houdement, C. (2007). Problem solving in France: Didactic and curricular perspectives. ZDM Int J Math Educ , 39 (5–6), 365–382. https://doi.org/10.1007/s11858-007-0048-x .

Article   Google Scholar  

Berger, W. (2014). A more beautiful question . Bloomsbury Publishing. Kindle Edition.

Brady, C., Ramírez, P., & Lesh, R. (2023). Problem posing and modeling: Confronting the dilemma of rigor or relevance. In T. L. Toh et al. (Eds.), Problem Posing and Problem Solving in Mathematics Education, pp: 33–50, Singapore: Springer. https://doi.org/10.1007/978-981-99-7205-0_3 .

Cai, J., & Hwang, S. (2023). Making mathematics challenging through problem posing in classroom. In R. Leikin (Ed.), Mathematical Challenges For All , Research in Mathematics Education, Springer: Switzerland, pp. 115–145, https://doi.org/10.1007/978-3-031-18868-8_7 .

Cai, J., & Rott, B. (2023). On understanding mathematical problem-posing processes. ZDM – Mathematics Education , 56 , 61–71. https://doi.org/10.1007/s11858-023-01536-w .

Cai, J., Hwang, S., & Melville, M. (2023). Mathematical problem-posing research: Thirty years of advances building on the publication of on mathematical problem solving. In J. Cai et al. (Eds.), Research Studies on Learning and Teaching of Mathematics, Research in Mathematics Education, Springer: Switzerland, pp: 1–25. https://doi.org/10.1007/978-3-031-35459-5_1 .

Cevikbas, M., & Kaiser, G. (2022). Can flipped classroom pedagogy offer promising perspectives for mathematics education on pan- demic-related issues? A systematic literature review. ZDM – Math- ematics Education . https://doi.org/10.1007/s11858-022-01388-w .

Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. Journal of Mathematical Behavior , 15 , 375–402.

Devlin, K. (2002). The millennium problems. The seven greatest unsolved mathematical puzzles of our time . Granta.

Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to support reasoning and sense making . National Council of Teachers of Mathematics, NCTM: Reston Va. ISBN 978-0-87353-641-7.

Ding, M., Wu, Y., Liu, Q., & Cai, J. (2022). Mathematics learning in Chinese contexts. ZDM -Mathematics Education , 54 , 577–496. https://doi.org/10.1007/s11858-022-01385-z .

Doorman, M., Drijvers, P., Dekker, T., Van den Heuvel- Panhuizen, M., de Lange, J., & Wijers, M. (2007). Problem solving as a challenge for mathematics education in the Netherlands. ZDM Int J Math Educ , 39 (5–6), 405–418. https://doi.org/10.1007/s11858-007-0043-2 .

Drijvers, P., & Sinclair, N. (2023). The role of digital technologies in mathematics education: Purposes and perspectives. ZDM-Mathematics Education . https://doi.org/10.1007/s11858-023-01535-x .

Engelbrecht, J., & Borba, M. C. (2023). Recent developments in using digital technology in mathematics education. ZDM -Mathematics Education . https://doi.org/10.1007/s11858-023-01530-2 .

Engelbrecht, J., Borba, M. C., & Kaiser, G. (2023). Will we ever teach mathematics again in the way we used to before the pandemic? ZDM– Mathematics Education , 55 , 1–16. https://doi.org/10.1007/s11858-022-01460-5 .

English, L. D. (2023). Ways of thinking in STEM-based problem solving. ZDM -Mathematics Education . https://doi.org/10.1007/s11858-023-01474-7 .

English, L. D., & Kirshner, D. (Eds.). (2016). Handbook of international research in mathematics education . NY. ISBN: 978-0-203-44894-6 (ebk). https://www.routledge.com/Handbook-of-International-Research-in-Mathematics-Education/English-Kirshner/p/book/9780415832045

Fried, M. N. (2014). Mathematics & mathematics education: Searching for common ground. In M.N. Fried, T. Dreyfus (Eds.), Mathematics & Mathematics Education: Searching for 3 Common Ground , Advances in Mathematics Education, pp: 3–22. https://doi.org/10.1007/978-94-007-7473-5_1 . NY: Springer.

Halmos, P. (1980). The heart of mathematics. American Mathematical Monthly , 87 (7), 519–524.

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., et al. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher , 25 (4), 12–21.

Hilbert, D. (1902). Mathematical problems. Bulletin of the American Mathematical Society , 8 , 437–479.

Højsted, I. H., Geranius, E., & Jankvist, U. T. (2022). Teachers’ facilitation of students’ mathematical reasoning in a dynamic geometry environment: An analysis through three lenses. In U. T. Jankvist, & E. Geraniou (Eds.), Mathematical competencies in the Digital era (pp. 271–292). Springer. https://doi.org/10.1007/978-3-031-10141-0_15 .

Kahneman, D. (2011). Thinking, fast and slow . Farrar, Straus and Giroux.

Koichu, B. (2014). Problem solving in mathematics and in mathematics education. In M.N. Fried, T. Dreyfus (Eds.), Mathematics & Mathematics Education: Searching for 113 Common Ground , Advances in Mathematics Education, pp: 113–135. Dordrecht: Springer. https://doi.org/10.1007/978-94-007-7473-5_8 .

Koichu, B., Cooper, J., & Widder, M. (2022). Implementation of problem solving in school: From intended to experienced. Implementation and Replication Studies in Mathematics Education , 2 (1), 76–106. https://doi.org/10.1163/26670127-bja10004 .

Krutestkii, V. A. (1976). The psychology of mathematical abilities in school children . University of Chicago Press, Chicago. ISBN: 0-226-45492-4.

Lee, N. H., Ng, W. L., & Lim, L. G. P. (2019). The intended school mathematics curriculum. In T. L. Toh et al. (Eds.), Mathematics Education in Singapore , Mathematics Education – An Asian Perspective, pp: 35–53. https://doi.org/10.1007/978-981-13-3573-0_3 .

Leikin, R., & Guberman, R. (2023). Creativity and challenge: Task complexity as a function of insight and multiplicity of solutions. R. Leikin (Ed.), Mathematical Challenges For All , Research in Mathematics Education, pp: 325–342. https://doi.org/10.1007/978-3-031-18868-8_17 .

Lester, F. K. Jr. (1994). Musing about mathematical problem-solving research: 1970–1994. Journal for Research in Mathematics Education , 25 (6), 660–675.

Lester, F. K. Jr. (2005). On the theoretical, conceptual, and philosophical foundation for research in mathematics education. Zdm Mathematics Education , 37 (6), 457–467. https://doi.org/10.1007/BF02655854 .

Lester, F. K. Jr., & Cai, J. (2016). Can mathematical problem solving be taught? Preliminary answers from 30 years of research. In P. Felmer, et al. (Eds.), Posing and solving Mathematical problems, Research in Mathematics Education (pp. 117–135). Springer. https://doi.org/10.1007/978-3-319-28023-3_8 .

Leung, A. (2011). An epistemic model of task design in dynamic geometry environment. Zdm , 43 , 325–336. https://doi.org/10.1007/s11858-011-0329-2 .

Leung, A., & Baccaglini-Frank, A. (Eds.). (2017). (Eds.). Digital Technologies in Designing Mathematics Education Tasks, Mathematics Education in the Digital Era 8, https://doi.org/10.1007/978-3-319-43423-0_1 .

Leung, A., & Bolite-Frant, J. (2015). Designing mathematics tasks: The role of tools. In A. Watson, & M. Ohtani (Eds.), Task design in mathematics education (pp. 191–225). New ICMI Study Series. https://doi.org/10.1007/978-3-319-09629-2_6 .

Liljedahl, P., & Cai, J. (2021). Empirical research on problem solving and problem pos- ing: A look at the state of the art. ZDM — Mathematics Education , 53 (4), 723–735. https://doi.org/10.1007/s11858-021-01291-w .

Pittalis, M., & Drijvers, P. (2023). Embodied instrumentation in a dynamic geometry environment: Eleven-year‐old students’ dragging schemes. Educational Studies in Mathematics , 113 , 181–205. https://doi.org/10.1007/s10649-023-10222-3 .

Pólya, G. (1945).; 2nd edition, 1957). How to solve it . Princeton University Press.

Rott, B., Specht, B., & Knipping, C. (2021). A descritive phase model of problem-solving processes. ZDM -Mathematics Education , 53 , 737–752. https://doi.org/10.1007/s11858-021-01244-3 .

Santos-Trigo, M. (2019). Mathematical Problem Solving and the use of digital technologies. In P. Liljedahl and M. Santos-Trigo (Eds.). Mathematical Problem Solving. ICME 13 Monographs , ISBN 978-3-030-10471-9, ISBN 978-3-030-10472-6 (eBook), Springer Nature Switzerland AG. Pp. 63–89 https://doi.org/10.1007/978-3-030-10472-6_4 .

Santos-Trigo, M. (2020a). Problem-solving in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 686–693). Springer. https://doi.org/10.1007/978-3-030-15789-0 .

Santos-Trigo, M. (2020b). Prospective and practicing teachers and the use of digital technologies in mathematical problem-solving approaches. In S. Llinares and O. Chapman (Eds.), International handbook of mathematics teacher education , vol 2, pp: 163–195. Boston: Brill Sense, ISBN 978-90-04-41896-7.

Santos-Trigo, M. (Ed.). (2023). Trends and developments of mathematical problem-solving research to update and support the use of digital technologies in post-confinement learning spaces. InT. L. Toh (Eds.), Problem Posing and Problem Solving in Mathematics Education , pp: 7–32. Springer Nature Singapore. https://doi.org/10.1007/978-981-99-7205-0_2 .

Santos-Trigo, M., & Reyes-Martínez, I. (2019). High school prospective teachers’ problem-solving reasoning that involves the coordinated use of digital technologies. International Journal of Mathematical Education in Science and Technology , 50 (2), 182–201. https://doi.org/10.1080/0020739X.2018.1489075 .

Santos-Trigo, M., Barrera-Mora, F., & Camacho-Machín, M. (2021). Teachers’ use of technology affordances to contextualize and dynamically enrich and extend mathematical problem-solving strategies. Mathematics , 9 (8), 793. https://doi.org/10.3390/math9080793 .

Santos-Trigo, M., Reyes-Martínez, I., & Gómez-Arciga, A. (2022). A conceptual framework to structure remote learning scenarios: A digital wall as a reflective tool for students to develop mathematics problem-solving competencies. Int J Learning Technology , 27–52. https://doi.org/10.1504/IJLT.2022.123686 .

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grows (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). Macmillan.

Schoenfeld, A. H. (2020). Mathematical practices, in theory and practice. ZDM Mathematics Education, 52, pp: 1163–1175. https://doi.org/10.1007/s11858-020-01162-w .

Schoenfeld, A. H. (2022). Why are learning and teaching mathematics so difficult? In M. Danesi (Ed.), Handbook of cognitive mathematics (pp. 1–35). Switzerland. https://doi.org/10.1007/978-3-030-44982-7_10-1%23DOI .

Schoenfeld, A. H. (2023). A theory of teaching. In A. K. Praetorius, & C. Y. Charalambous (Eds.), Theorizing teaching (pp. 159–187). Springer. https://doi.org/10.1007/978-3-031-25613-4_6 .

Sinclair, N., & Ferrara, F. (2023). Towards a Socio-material Reframing of Mathematically Challenging Tasks. In R. Leikin (Ed.), Mathematical Challenges For All , Research in Mathematics Education, pp: 307–323. https://doi.org/10.1007/978-3-031-18868-8_16 .

Stake, R. E. (2000). Case studies. In N. K. Denzin, & Y. S. Lincoln (Eds.), Handbook of qualitative research (pp. 435–454). Sage.

Thurston, P. W. (1994). On proof and progress in mathematics. Bull Amer Math Soc , 30 (2), 161–177.

Toh, T. L., Santos-Trigo, M., Chua, P. H., Abdullah, N. A., & Zhang, D. (Eds.). (2023). Problem posing and problem solving in mathematics education: Internationa research and practice trends . Springer Nature Singpore. https://doi.org/10.1007/978-981-99-7205-0 .

Törner, G., Schoenfeld, A. H., & Reiss, K. M. (Eds.). (2007). Problem solving around the world: Summing up the state of the art [Special Issue]. ZDM — Mathematics Education , 39 (5–6). https://doi.org/10.1007/s11858-007-0053-0 .

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Santos-Trigo, M. Problem solving in mathematics education: tracing its foundations and current research-practice trends. ZDM Mathematics Education (2024). https://doi.org/10.1007/s11858-024-01578-8

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