problem solving use in teaching

Center for Teaching

Teaching problem solving.

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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-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.

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

Why Every Educator Needs to Teach Problem-Solving Skills

Strong problem-solving skills will help students be more resilient and will increase their academic and career success .

Want to learn more about how to measure and teach students’ higher-order skills, including problem solving, critical thinking, and written communication?

Problem-solving skills are essential in school, careers, and life.

Problem-solving skills are important for every student to master. They help individuals navigate everyday life and find solutions to complex issues and challenges. These skills are especially valuable in the workplace, where employees are often required to solve problems and make decisions quickly and effectively.

Problem-solving skills are also needed for students’ personal growth and development because they help individuals overcome obstacles and achieve their goals. By developing strong problem-solving skills, students can improve their overall quality of life and become more successful in their personal and professional endeavors.

Problem-Solving Skills Help Students…

develop resilience.

Problem-solving skills are an integral part of resilience and the ability to persevere through challenges and adversity. To effectively work through and solve a problem, students must be able to think critically and creatively. Critical and creative thinking help students approach a problem objectively, analyze its components, and determine different ways to go about finding a solution.

This process in turn helps students build self-efficacy . When students are able to analyze and solve a problem, this increases their confidence, and they begin to realize the power they have to advocate for themselves and make meaningful change.

When students gain confidence in their ability to work through problems and attain their goals, they also begin to build a growth mindset . According to leading resilience researcher, Carol Dweck, “in a growth mindset, people believe that their most basic abilities can be developed through dedication and hard work—brains and talent are just the starting point. This view creates a love of learning and a resilience that is essential for great accomplishment.”

Set and Achieve Goals

Students who possess strong problem-solving skills are better equipped to set and achieve their goals. By learning how to identify problems, think critically, and develop solutions, students can become more self-sufficient and confident in their ability to achieve their goals. Additionally, problem-solving skills are used in virtually all fields, disciplines, and career paths, which makes them important for everyone. Building strong problem-solving skills will help students enhance their academic and career performance and become more competitive as they begin to seek full-time employment after graduation or pursue additional education and training.

Resolve Conflicts

In addition to increased social and emotional skills like self-efficacy and goal-setting, problem-solving skills teach students how to cooperate with others and work through disagreements and conflicts. Problem-solving promotes “thinking outside the box” and approaching a conflict by searching for different solutions. This is a very different (and more effective!) method than a more stagnant approach that focuses on placing blame or getting stuck on elements of a situation that can’t be changed.

While it’s natural to get frustrated or feel stuck when working through a conflict, students with strong problem-solving skills will be able to work through these obstacles, think more rationally, and address the situation with a more solution-oriented approach. These skills will be valuable for students in school, their careers, and throughout their lives.

Achieve Success

We are all faced with problems every day. Problems arise in our personal lives, in school and in our jobs, and in our interactions with others. Employers especially are looking for candidates with strong problem-solving skills. In today’s job market, most jobs require the ability to analyze and effectively resolve complex issues. Students with strong problem-solving skills will stand out from other applicants and will have a more desirable skill set.

In a recent opinion piece published by The Hechinger Report , Virgel Hammonds, Chief Learning Officer at KnowledgeWorks, stated “Our world presents increasingly complex challenges. Education must adapt so that it nurtures problem solvers and critical thinkers.” Yet, the “traditional K–12 education system leaves little room for students to engage in real-world problem-solving scenarios.” This is the reason that a growing number of K–12 school districts and higher education institutions are transforming their instructional approach to personalized and competency-based learning, which encourage students to make decisions, problem solve and think critically as they take ownership of and direct their educational journey.

Problem-Solving Skills Can Be Measured and Taught

Research shows that problem-solving skills can be measured and taught. One effective method is through performance-based assessments which require students to demonstrate or apply their knowledge and higher-order skills to create a response or product or do a task.

What Are Performance-Based Assessments?

With the No Child Left Behind Act (2002), the use of standardized testing became the primary way to measure student learning in the U.S. The legislative requirements of this act shifted the emphasis to standardized testing, and this led to a decline in nontraditional testing methods .

But many educators, policy makers, and parents have concerns with standardized tests. Some of the top issues include that they don’t provide feedback on how students can perform better, they don’t value creativity, they are not representative of diverse populations, and they can be disadvantageous to lower-income students.

While standardized tests are still the norm, U.S. Secretary of Education Miguel Cardona is encouraging states and districts to move away from traditional multiple choice and short response tests and instead use performance-based assessment, competency-based assessments, and other more authentic methods of measuring students abilities and skills rather than rote learning.

Performance-based assessments measure whether students can apply the skills and knowledge learned from a unit of study. Typically, a performance task challenges students to use their higher-order skills to complete a project or process. Tasks can range from an essay to a complex proposal or design.

Preview a Performance-Based Assessment

Want a closer look at how performance-based assessments work? Preview CAE’s K–12 and Higher Education assessments and see how CAE’s tools help students develop critical thinking, problem-solving, and written communication skills.

Performance-Based Assessments Help Students Build and Practice Problem-Solving Skills

In addition to effectively measuring students’ higher-order skills, including their problem-solving skills, performance-based assessments can help students practice and build these skills. Through the assessment process, students are given opportunities to practically apply their knowledge in real-world situations. By demonstrating their understanding of a topic, students are required to put what they’ve learned into practice through activities such as presentations, experiments, and simulations.

This type of problem-solving assessment tool requires students to analyze information and choose how to approach the presented problems. This process enhances their critical thinking skills and creativity, as well as their problem-solving skills. Unlike traditional assessments based on memorization or reciting facts, performance-based assessments focus on the students’ decisions and solutions, and through these tasks students learn to bridge the gap between theory and practice.

Performance-based assessments like CAE’s College and Career Readiness Assessment (CRA+) and Collegiate Learning Assessment (CLA+) provide students with in-depth reports that show them which higher-order skills they are strongest in and which they should continue to develop. This feedback helps students and their teachers plan instruction and supports to deepen their learning and improve their mastery of critical skills.

Explore CAE’s Problem-Solving Assessments

CAE offers performance-based assessments that measure student proficiency in higher-order skills including problem solving, critical thinking, and written communication.

College and Career Readiness Assessment (CCRA+) for secondary education and
Collegiate Learning Assessment (CLA+) for higher education.

Our solution also includes instructional materials, practice models, and professional development.

We can help you create a program to build students’ problem-solving skills that includes:

Measuring students’ problem-solving skills through a performance-based assessment
Using the problem-solving assessment data to inform instruction and tailor interventions
Teaching students problem-solving skills and providing practice opportunities in real-life scenarios
Supporting educators with quality professional development

Get started with our problem-solving assessment tools to measure and build students’ problem-solving skills today! These skills will be invaluable to students now and in the future.

Ready to Get Started?

Learn more about cae’s suite of products and let’s get started measuring and teaching students important higher-order skills like problem solving..

Teaching Problem-Solving Skills

Many instructors design opportunities for students to solve “problems”. But are their students solving true problems or merely participating in practice exercises? The former stresses critical thinking and decision making skills whereas the latter requires only the application of previously learned procedures.

Problem solving is often broadly defined as "the ability to understand the environment, identify complex problems, review related information to develop, evaluate strategies and implement solutions to build the desired outcome" (Fissore, C. et al, 2021). True problem solving is the process of applying a method – not known in advance – to a problem that is subject to a specific set of conditions and that the problem solver has not seen before, in order to obtain a satisfactory solution.

Below you will find some basic principles for teaching problem solving and one model to implement in your classroom teaching.

Principles for teaching problem solving

Model a useful problem-solving method . Problem solving can be difficult and sometimes tedious. Show students how to be patient and persistent, and how to follow a structured method, such as Woods’ model described below. Articulate your method as you use it so students see the connections.
Teach within a specific context . Teach problem-solving skills in the context in which they will be used by students (e.g., mole fraction calculations in a chemistry course). Use real-life problems in explanations, examples, and exams. Do not teach problem solving as an independent, abstract skill.
Help students understand the problem . In order to solve problems, students need to define the end goal. This step is crucial to successful learning of problem-solving skills. If you succeed at helping students answer the questions “what?” and “why?”, finding the answer to “how?” will be easier.
Take enough time . When planning a lecture/tutorial, budget enough time for: understanding the problem and defining the goal (both individually and as a class); dealing with questions from you and your students; making, finding, and fixing mistakes; and solving entire problems in a single session.
Ask questions and make suggestions . Ask students to predict “what would happen if …” or explain why something happened. This will help them to develop analytical and deductive thinking skills. Also, ask questions and make suggestions about strategies to encourage students to reflect on the problem-solving strategies that they use.
Link errors to misconceptions . Use errors as evidence of misconceptions, not carelessness or random guessing. Make an effort to isolate the misconception and correct it, then teach students to do this by themselves. We can all learn from mistakes.

Woods’ problem-solving model

Define the problem.

The system . Have students identify the system under study (e.g., a metal bridge subject to certain forces) by interpreting the information provided in the problem statement. Drawing a diagram is a great way to do this.
Known(s) and concepts . List what is known about the problem, and identify the knowledge needed to understand (and eventually) solve it.
Unknown(s) . Once you have a list of knowns, identifying the unknown(s) becomes simpler. One unknown is generally the answer to the problem, but there may be other unknowns. Be sure that students understand what they are expected to find.
Units and symbols . One key aspect in problem solving is teaching students how to select, interpret, and use units and symbols. Emphasize the use of units whenever applicable. Develop a habit of using appropriate units and symbols yourself at all times.
Constraints . All problems have some stated or implied constraints. Teach students to look for the words "only", "must", "neglect", or "assume" to help identify the constraints.
Criteria for success . Help students consider, from the beginning, what a logical type of answer would be. What characteristics will it possess? For example, a quantitative problem will require an answer in some form of numerical units (e.g., $/kg product, square cm, etc.) while an optimization problem requires an answer in the form of either a numerical maximum or minimum.

Think about it

“Let it simmer”. Use this stage to ponder the problem. Ideally, students will develop a mental image of the problem at hand during this stage.
Identify specific pieces of knowledge . Students need to determine by themselves the required background knowledge from illustrations, examples and problems covered in the course.
Collect information . Encourage students to collect pertinent information such as conversion factors, constants, and tables needed to solve the problem.

Plan a solution

Consider possible strategies . Often, the type of solution will be determined by the type of problem. Some common problem-solving strategies are: compute; simplify; use an equation; make a model, diagram, table, or chart; or work backwards.
Choose the best strategy . Help students to choose the best strategy by reminding them again what they are required to find or calculate.

Carry out the plan

Be patient . Most problems are not solved quickly or on the first attempt. In other cases, executing the solution may be the easiest step.
Be persistent . If a plan does not work immediately, do not let students get discouraged. Encourage them to try a different strategy and keep trying.

Encourage students to reflect. Once a solution has been reached, students should ask themselves the following questions:

Does the answer make sense?
Does it fit with the criteria established in step 1?
Did I answer the question(s)?
What did I learn by doing this?
Could I have done the problem another way?

If you would like support applying these tips to your own teaching, CTE staff members are here to help. View the CTE Support page to find the most relevant staff member to contact.

Fissore, C., Marchisio, M., Roman, F., & Sacchet, M. (2021). Development of problem solving skills with Maple in higher education. In: Corless, R.M., Gerhard, J., Kotsireas, I.S. (eds) Maple in Mathematics Education and Research. MC 2020. Communications in Computer and Information Science, vol 1414. Springer, Cham. https://doi.org/10.1007/978-3-030-81698-8_15
Foshay, R., & Kirkley, J. (1998). Principles for Teaching Problem Solving. TRO Learning Inc., Edina MN. (PDF) Principles for Teaching Problem Solving (researchgate.net)
Hayes, J.R. (1989). The Complete Problem Solver. 2nd Edition. Hillsdale, NJ: Lawrence Erlbaum Associates.
Woods, D.R., Wright, J.D., Hoffman, T.W., Swartman, R.K., Doig, I.D. (1975). Teaching Problem solving Skills.
Engineering Education. Vol 1, No. 1. p. 238. Washington, DC: The American Society for Engineering Education.

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Don’t Just Tell Students to Solve Problems. Teach Them How.

The positive impact of an innovative UC San Diego problem-solving educational curriculum continues to grow

Daniel Kane - [email protected]

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Not getting stuck

One of the overarching goals of this project is to teach both problem-identification and problem-solving skills that help students avoid getting stuck during the learning process. Stuck feelings lead to frustration – and when it’s a Science, Technology, Engineering and Math (STEM) project, that frustration can lead students to feel they don’t belong in a STEM major or a STEM career. Instead, the UC San Diego curriculum is designed to give students the tools that lead to reactions like “this class is hard, but I know I can do this!” – as Ogo, a celebrated high school biomedical sciences and technology teacher, put it.

Three years into the curriculum development effort, the light-hearted energy of the students combined with their intense focus points to success. On the last day of the class, Mourad Mjahed PhD, Director of the MESA Program at Southwestern College’s School of Mathematics, Science and Engineering came to UC San Diego to see the final project presentations made by his 22 MESA students.

“Industry is looking for students who have learned from their failures and who have worked outside of their comfort zones,” said Mjahed. The UC San Diego problem-solving curriculum, Mjahed noted, is an opportunity for students to build the skills and the confidence to learn from their failures and to work outside their comfort zone. “And from there, they see pathways to real careers,” he said.

What does it mean to explicitly teach problem solving?

This approach to teaching problem solving includes a significant focus on learning to identify the problem that actually needs to be solved, in order to avoid solving the wrong problem. The curriculum is organized so that each day is a complete experience. It begins with the teacher introducing the problem-identification or problem-solving strategy of the day. The teacher then presents case studies of that particular strategy in action. Next, the students get introduced to the day’s challenge project. Working in teams, the students compete to win the challenge while integrating the day’s technique. Finally, the class reconvenes to reflect. They discuss what worked and didn't work with their designs as well as how they could have used the day’s problem-identification or problem-solving technique more effectively.

The challenges are designed to be engaging – and over three years, they have been refined to be even more engaging. But the student engagement is about much more than being entertained. Many of the students recognize early on that the problem-identification and problem-solving skills they are learning can be applied not just in the classroom, but in other classes and in life in general.

Gabriel from Southwestern College is one of the students who saw benefits outside the classroom almost immediately. In addition to taking the UC San Diego problem-solving course, Gabriel was concurrently enrolled in an online computer science programming class. He said he immediately started applying the UC San Diego problem-identification and troubleshooting strategies to his coding assignments.

Gabriel noted that he was given a coding-specific troubleshooting strategy in the computer science course, but the more general problem-identification strategies from the UC San Diego class had been extremely helpful. It’s critical to “find the right problem so you can get the right solution. The strategies here,” he said, “they work everywhere.”

Phan echoed this sentiment. “We believe this curriculum can prepare students for the technical workforce. It can prepare students to be impactful for any career path.”

The goal is to be able to offer the course in community colleges for course credit that transfers to the UC, and to possibly offer a version of the course to incoming students at UC San Diego.

As the team continues to work towards integrating the curriculum in both standardized high school courses such as physics, and incorporating the content as a part of the general education curriculum at UC San Diego, the project is expected to impact thousands more students across San Diego annually.

Portrait of the Problem-Solving Curriculum

On a sunny Wednesday in July 2023, an experiential-learning classroom was full of San Diego community college students. They were about half-way through the three-week problem-solving course at UC San Diego, held in the campus’ EnVision Arts and Engineering Maker Studio. On this day, the students were challenged to build a contraption that would propel at least six ping pong balls along a kite string spanning the laboratory. The only propulsive force they could rely on was the air shooting out of a party balloon.

A team of three students from Southwestern College – Valeria, Melissa and Alondra – took an early lead in the classroom competition. They were the first to use a plastic bag instead of disposable cups to hold the ping pong balls. Using a bag, their design got more than half-way to the finish line – better than any other team at the time – but there was more work to do.

As the trio considered what design changes to make next, they returned to the problem-solving theme of the day: unintended consequences. Earlier in the day, all the students had been challenged to consider unintended consequences and ask questions like: When you design to reduce friction, what happens? Do new problems emerge? Did other things improve that you hadn’t anticipated?

Other groups soon followed Valeria, Melissa and Alondra’s lead and began iterating on their own plastic-bag solutions to the day’s challenge. New unintended consequences popped up everywhere. Switching from cups to a bag, for example, reduced friction but sometimes increased wind drag.

Over the course of several iterations, Valeria, Melissa and Alondra made their bag smaller, blew their balloon up bigger, and switched to a different kind of tape to get a better connection with the plastic straw that slid along the kite string, carrying the ping pong balls.

One of the groups on the other side of the room watched the emergence of the plastic-bag solution with great interest.

“We tried everything, then we saw a team using a bag,” said Alexander, a student from City College. His team adopted the plastic-bag strategy as well, and iterated on it like everyone else. They also chose to blow up their balloon with a hand pump after the balloon was already attached to the bag filled with ping pong balls – which was unique.

“I don’t want to be trying to put the balloon in place when it's about to explode,” Alexander explained.

Asked about whether the structured problem solving approaches were useful, Alexander’s teammate Brianna, who is a Southwestern College student, talked about how the problem-solving tools have helped her get over mental blocks. “Sometimes we make the most ridiculous things work,” she said. “It’s a pretty fun class for sure.”

Yoshadara, a City College student who is the third member of this team, described some of the problem solving techniques this way: “It’s about letting yourself be a little absurd.”

Alexander jumped back into the conversation. “The value is in the abstraction. As students, we learn to look at the problem solving that worked and then abstract out the problem solving strategy that can then be applied to other challenges. That’s what mathematicians do all the time,” he said, adding that he is already thinking about how he can apply the process of looking at unintended consequences to improve both how he plays chess and how he goes about solving math problems.

Looking ahead, the goal is to empower as many students as possible in the San Diego area and beyond to learn to problem solve more enjoyably. It’s a concrete way to give students tools that could encourage them to thrive in the growing number of technical careers that require sharp problem-solving skills, whether or not they require a four-year degree.

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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. 

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Teaching problem solving

Strategies for teaching problem solving apply across disciplines and instructional contexts. First, introduce the problem and explain how people in your discipline generally make sense of the given information. Then, explain how to apply these approaches to solve the problem.

Introducing the problem

Explaining how people in your discipline understand and interpret these types of problems can help students develop the skills they need to understand the problem (and find a solution). After introducing how you would go about solving a problem, you could then ask students to:

frame the problem in their own words
define key terms and concepts
determine statements that accurately represent the givens of a problem
identify analogous problems
determine what information is needed to solve the problem

Working on solutions

In the solution phase, one develops and then implements a coherent plan for solving the problem. As you help students with this phase, you might ask them to:

identify the general model or procedure they have in mind for solving the problem
set sub-goals for solving the problem
identify necessary operations and steps
draw conclusions
carry out necessary operations

You can help students tackle a problem effectively by asking them to:

systematically explain each step and its rationale
explain how they would approach solving the problem
help you solve the problem by posing questions at key points in the process
work together in small groups (3 to 5 students) to solve the problem and then have the solution presented to the rest of the class (either by you or by a student in the group)

In all cases, the more you get the students to articulate their own understandings of the problem and potential solutions, the more you can help them develop their expertise in approaching problems in your discipline.

Teaching problem solving: Let students get ‘stuck’ and ‘unstuck’

Subscribe to the center for universal education bulletin, kate mills and km kate mills literacy interventionist - red bank primary school helyn kim helyn kim former brookings expert.

October 31, 2017

This is the second in a six-part blog series on teaching 21st century skills , including problem solving , metacognition , critical thinking , and collaboration , in classrooms.

In the real world, students encounter problems that are complex, not well defined, and lack a clear solution and approach. They need to be able to identify and apply different strategies to solve these problems. However, problem solving skills do not necessarily develop naturally; they need to be explicitly taught in a way that can be transferred across multiple settings and contexts.

Here’s what Kate Mills, who taught 4 th grade for 10 years at Knollwood School in New Jersey and is now a Literacy Interventionist at Red Bank Primary School, has to say about creating a classroom culture of problem solvers:

Helping my students grow to be people who will be successful outside of the classroom is equally as important as teaching the curriculum. From the first day of school, I intentionally choose language and activities that help to create a classroom culture of problem solvers. I want to produce students who are able to think about achieving a particular goal and manage their mental processes . This is known as metacognition , and research shows that metacognitive skills help students become better problem solvers.

I begin by “normalizing trouble” in the classroom. Peter H. Johnston teaches the importance of normalizing struggle , of naming it, acknowledging it, and calling it what it is: a sign that we’re growing. The goal is for the students to accept challenge and failure as a chance to grow and do better.

I look for every chance to share problems and highlight how the students— not the teachers— worked through those problems. There is, of course, coaching along the way. For example, a science class that is arguing over whose turn it is to build a vehicle will most likely need a teacher to help them find a way to the balance the work in an equitable way. Afterwards, I make it a point to turn it back to the class and say, “Do you see how you …” By naming what it is they did to solve the problem , students can be more independent and productive as they apply and adapt their thinking when engaging in future complex tasks.

After a few weeks, most of the class understands that the teachers aren’t there to solve problems for the students, but to support them in solving the problems themselves. With that important part of our classroom culture established, we can move to focusing on the strategies that students might need.

Here’s one way I do this in the classroom:

I show the broken escalator video to the class. Since my students are fourth graders, they think it’s hilarious and immediately start exclaiming, “Just get off! Walk!”

When the video is over, I say, “Many of us, probably all of us, are like the man in the video yelling for help when we get stuck. When we get stuck, we stop and immediately say ‘Help!’ instead of embracing the challenge and trying new ways to work through it.” I often introduce this lesson during math class, but it can apply to any area of our lives, and I can refer to the experience and conversation we had during any part of our day.

Research shows that just because students know the strategies does not mean they will engage in the appropriate strategies. Therefore, I try to provide opportunities where students can explicitly practice learning how, when, and why to use which strategies effectively so that they can become self-directed learners.

For example, I give students a math problem that will make many of them feel “stuck”. I will say, “Your job is to get yourselves stuck—or to allow yourselves to get stuck on this problem—and then work through it, being mindful of how you’re getting yourselves unstuck.” As students work, I check-in to help them name their process: “How did you get yourself unstuck?” or “What was your first step? What are you doing now? What might you try next?” As students talk about their process, I’ll add to a list of strategies that students are using and, if they are struggling, help students name a specific process. For instance, if a student says he wrote the information from the math problem down and points to a chart, I will say: “Oh that’s interesting. You pulled the important information from the problem out and organized it into a chart.” In this way, I am giving him the language to match what he did, so that he now has a strategy he could use in other times of struggle.

The charts grow with us over time and are something that we refer to when students are stuck or struggling. They become a resource for students and a way for them to talk about their process when they are reflecting on and monitoring what did or did not work.

For me, as a teacher, it is important that I create a classroom environment in which students are problem solvers. This helps tie struggles to strategies so that the students will not only see value in working harder but in working smarter by trying new and different strategies and revising their process. In doing so, they will more successful the next time around.

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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Science, technology and society

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 .

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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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Critical thinking and problem-solving, jump to: , what is critical thinking, characteristics of critical thinking, why teach critical thinking.

Teaching Strategies to Help Promote Critical Thinking Skills

References and Resources

When examining the vast literature on critical thinking, various definitions of critical thinking emerge. Here are some samples:

"Critical thinking is the intellectually disciplined process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and/or evaluating information gathered from, or generated by, observation, experience, reflection, reasoning, or communication, as a guide to belief and action" (Scriven, 1996).
"Most formal definitions characterize critical thinking as the intentional application of rational, higher order thinking skills, such as analysis, synthesis, problem recognition and problem solving, inference, and evaluation" (Angelo, 1995, p. 6).
"Critical thinking is thinking that assesses itself" (Center for Critical Thinking, 1996b).
"Critical thinking is the ability to think about one's thinking in such a way as 1. To recognize its strengths and weaknesses and, as a result, 2. To recast the thinking in improved form" (Center for Critical Thinking, 1996c).

Perhaps the simplest definition is offered by Beyer (1995) : "Critical thinking... means making reasoned judgments" (p. 8). Basically, Beyer sees critical thinking as using criteria to judge the quality of something, from cooking to a conclusion of a research paper. In essence, critical thinking is a disciplined manner of thought that a person uses to assess the validity of something (statements, news stories, arguments, research, etc.).

Back

Wade (1995) identifies eight characteristics of critical thinking. Critical thinking involves asking questions, defining a problem, examining evidence, analyzing assumptions and biases, avoiding emotional reasoning, avoiding oversimplification, considering other interpretations, and tolerating ambiguity. Dealing with ambiguity is also seen by Strohm & Baukus (1995) as an essential part of critical thinking, "Ambiguity and doubt serve a critical-thinking function and are a necessary and even a productive part of the process" (p. 56).

Another characteristic of critical thinking identified by many sources is metacognition. Metacognition is thinking about one's own thinking. More specifically, "metacognition is being aware of one's thinking as one performs specific tasks and then using this awareness to control what one is doing" (Jones & Ratcliff, 1993, p. 10 ).

In the book, Critical Thinking, Beyer elaborately explains what he sees as essential aspects of critical thinking. These are:

Dispositions: Critical thinkers are skeptical, open-minded, value fair-mindedness, respect evidence and reasoning, respect clarity and precision, look at different points of view, and will change positions when reason leads them to do so.
Criteria: To think critically, must apply criteria. Need to have conditions that must be met for something to be judged as believable. Although the argument can be made that each subject area has different criteria, some standards apply to all subjects. "... an assertion must... be based on relevant, accurate facts; based on credible sources; precise; unbiased; free from logical fallacies; logically consistent; and strongly reasoned" (p. 12).
Argument: Is a statement or proposition with supporting evidence. Critical thinking involves identifying, evaluating, and constructing arguments.
Reasoning: The ability to infer a conclusion from one or multiple premises. To do so requires examining logical relationships among statements or data.
Point of View: The way one views the world, which shapes one's construction of meaning. In a search for understanding, critical thinkers view phenomena from many different points of view.
Procedures for Applying Criteria: Other types of thinking use a general procedure. Critical thinking makes use of many procedures. These procedures include asking questions, making judgments, and identifying assumptions.

Oliver & Utermohlen (1995) see students as too often being passive receptors of information. Through technology, the amount of information available today is massive. This information explosion is likely to continue in the future. Students need a guide to weed through the information and not just passively accept it. Students need to "develop and effectively apply critical thinking skills to their academic studies, to the complex problems that they will face, and to the critical choices they will be forced to make as a result of the information explosion and other rapid technological changes" (Oliver & Utermohlen, p. 1 ).

As mentioned in the section, Characteristics of Critical Thinking , critical thinking involves questioning. It is important to teach students how to ask good questions, to think critically, in order to continue the advancement of the very fields we are teaching. "Every field stays alive only to the extent that fresh questions are generated and taken seriously" (Center for Critical Thinking, 1996a ).

Beyer sees the teaching of critical thinking as important to the very state of our nation. He argues that to live successfully in a democracy, people must be able to think critically in order to make sound decisions about personal and civic affairs. If students learn to think critically, then they can use good thinking as the guide by which they live their lives.

Teaching Strategies to Help Promote Critical Thinking

The 1995, Volume 22, issue 1, of the journal, Teaching of Psychology , is devoted to the teaching critical thinking. Most of the strategies included in this section come from the various articles that compose this issue.

CATS (Classroom Assessment Techniques): Angelo stresses the use of ongoing classroom assessment as a way to monitor and facilitate students' critical thinking. An example of a CAT is to ask students to write a "Minute Paper" responding to questions such as "What was the most important thing you learned in today's class? What question related to this session remains uppermost in your mind?" The teacher selects some of the papers and prepares responses for the next class meeting.
Cooperative Learning Strategies: Cooper (1995) argues that putting students in group learning situations is the best way to foster critical thinking. "In properly structured cooperative learning environments, students perform more of the active, critical thinking with continuous support and feedback from other students and the teacher" (p. 8).
Case Study /Discussion Method: McDade (1995) describes this method as the teacher presenting a case (or story) to the class without a conclusion. Using prepared questions, the teacher then leads students through a discussion, allowing students to construct a conclusion for the case.
Using Questions: King (1995) identifies ways of using questions in the classroom:
Reciprocal Peer Questioning: Following lecture, the teacher displays a list of question stems (such as, "What are the strengths and weaknesses of...). Students must write questions about the lecture material. In small groups, the students ask each other the questions. Then, the whole class discusses some of the questions from each small group.
Reader's Questions: Require students to write questions on assigned reading and turn them in at the beginning of class. Select a few of the questions as the impetus for class discussion.
Conference Style Learning: The teacher does not "teach" the class in the sense of lecturing. The teacher is a facilitator of a conference. Students must thoroughly read all required material before class. Assigned readings should be in the zone of proximal development. That is, readings should be able to be understood by students, but also challenging. The class consists of the students asking questions of each other and discussing these questions. The teacher does not remain passive, but rather, helps "direct and mold discussions by posing strategic questions and helping students build on each others' ideas" (Underwood & Wald, 1995, p. 18 ).
Use Writing Assignments: Wade sees the use of writing as fundamental to developing critical thinking skills. "With written assignments, an instructor can encourage the development of dialectic reasoning by requiring students to argue both [or more] sides of an issue" (p. 24).
Written dialogues: Give students written dialogues to analyze. In small groups, students must identify the different viewpoints of each participant in the dialogue. Must look for biases, presence or exclusion of important evidence, alternative interpretations, misstatement of facts, and errors in reasoning. Each group must decide which view is the most reasonable. After coming to a conclusion, each group acts out their dialogue and explains their analysis of it.
Spontaneous Group Dialogue: One group of students are assigned roles to play in a discussion (such as leader, information giver, opinion seeker, and disagreer). Four observer groups are formed with the functions of determining what roles are being played by whom, identifying biases and errors in thinking, evaluating reasoning skills, and examining ethical implications of the content.
Ambiguity: Strohm & Baukus advocate producing much ambiguity in the classroom. Don't give students clear cut material. Give them conflicting information that they must think their way through.
Angelo, T. A. (1995). Beginning the dialogue: Thoughts on promoting critical thinking: Classroom assessment for critical thinking. Teaching of Psychology, 22(1), 6-7.
Beyer, B. K. (1995). Critical thinking. Bloomington, IN: Phi Delta Kappa Educational Foundation.
Center for Critical Thinking (1996a). The role of questions in thinking, teaching, and learning. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
Center for Critical Thinking (1996b). Structures for student self-assessment. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univclass/trc.nclk
Center for Critical Thinking (1996c). Three definitions of critical thinking [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
Cooper, J. L. (1995). Cooperative learning and critical thinking. Teaching of Psychology, 22(1), 7-8.
Jones, E. A. & Ratcliff, G. (1993). Critical thinking skills for college students. National Center on Postsecondary Teaching, Learning, and Assessment, University Park, PA. (Eric Document Reproduction Services No. ED 358 772)
King, A. (1995). Designing the instructional process to enhance critical thinking across the curriculum: Inquiring minds really do want to know: Using questioning to teach critical thinking. Teaching of Psychology, 22 (1) , 13-17.
McDade, S. A. (1995). Case study pedagogy to advance critical thinking. Teaching Psychology, 22(1), 9-10.
Oliver, H. & Utermohlen, R. (1995). An innovative teaching strategy: Using critical thinking to give students a guide to the future.(Eric Document Reproduction Services No. 389 702)
Robertson, J. F. & Rane-Szostak, D. (1996). Using dialogues to develop critical thinking skills: A practical approach. Journal of Adolescent & Adult Literacy, 39(7), 552-556.
Scriven, M. & Paul, R. (1996). Defining critical thinking: A draft statement for the National Council for Excellence in Critical Thinking. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
Strohm, S. M., & Baukus, R. A. (1995). Strategies for fostering critical thinking skills. Journalism and Mass Communication Educator, 50 (1), 55-62.
Underwood, M. K., & Wald, R. L. (1995). Conference-style learning: A method for fostering critical thinking with heart. Teaching Psychology, 22(1), 17-21.
Wade, C. (1995). Using writing to develop and assess critical thinking. Teaching of Psychology, 22(1), 24-28.

On the Internet

Carr, K. S. (1990). How can we teach critical thinking. Eric Digest. [On-line]. Available HTTP: http://ericps.ed.uiuc.edu/eece/pubs/digests/1990/carr90.html
The Center for Critical Thinking (1996). Home Page. Available HTTP: http://www.criticalthinking.org/University/
Ennis, Bob (No date). Critical thinking. [On-line], April 4, 1997. Available HTTP: http://www.cof.orst.edu/cof/teach/for442/ct.htm
Montclair State University (1995). Curriculum resource center. Critical thinking resources: An annotated bibliography. [On-line]. Available HTTP: http://www.montclair.edu/Pages/CRC/Bibliographies/CriticalThinking.html
No author, No date. Critical Thinking is ... [On-line], April 4, 1997. Available HTTP: http://library.usask.ca/ustudy/critical/
Sheridan, Marcia (No date). Internet education topics hotlink page. [On-line], April 4, 1997. Available HTTP: http://sun1.iusb.edu/~msherida/topics/critical.html

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Problem based learning: a teacher's guide

December 10, 2021

Find out how teachers use problem-based learning models to improve engagement and drive attainment.

Main, P (2021, December 10). Problem based learning: a teacher's guide. Retrieved from https://www.structural-learning.com/post/problem-based-learning-a-teachers-guide

What is problem-based learning?

Problem-based learning (PBL) is a style of teaching that encourages students to become the drivers of their learning process . Problem-based learning involves complex learning issues from real-world problems and makes them the classroom's topic of discussion ; encouraging students to understand concepts through problem-solving skills rather than simply learning facts. When schools find time in the curriculum for this style of teaching it offers students an authentic vehicle for the integration of knowledge .

Embracing this pedagogical approach enables schools to balance subject knowledge acquisition with a skills agenda . Often used in medical education, this approach has equal significance in mainstream education where pupils can apply their knowledge to real-life problems.

PBL is not only helpful in learning course content , but it can also promote the development of problem-solving abilities , critical thinking skills , and communication skills while providing opportunities to work in groups , find and analyse research materials , and take part in life-long learning .

PBL is a student-centred teaching method in which students understand a topic by working in groups. They work out an open-ended problem , which drives the motivation to learn. These sorts of theories of teaching do require schools to invest time and resources into supporting self-directed learning. Not all curriculum knowledge is best acquired through this process, rote learning still has its place in certain situations. In this article, we will look at how we can equip our students to take more ownership of the learning process and utilise more sophisticated ways for the integration of knowledge .

Philosophical Underpinnings of PBL

Problem-Based Learning (PBL), with its roots in the philosophies of John Dewey, Maria Montessori, and Jerome Bruner, aligns closely with the social constructionist view of learning. This approach positions learners as active participants in the construction of knowledge, contrasting with traditional models of instruction where learners are seen as passive recipients of information.

Dewey, a seminal figure in progressive education, advocated for active learning and real-world problem-solving, asserting that learning is grounded in experience and interaction. In PBL, learners tackle complex, real-world problems, which mirrors Dewey's belief in the interconnectedness of education and practical life.

Montessori also endorsed learner-centric, self-directed learning, emphasizing the child's potential to construct their own learning experiences. This parallels with PBL’s emphasis on self-directed learning, where students take ownership of their learning process.

Jerome Bruner’s theories underscored the idea of learning as an active, social process. His concept of a 'spiral curriculum' – where learning is revisited in increasing complexity – can be seen reflected in the iterative problem-solving process in PBL.

Webb’s Depth of Knowledge (DOK) framework aligns with PBL as it encourages higher-order cognitive skills. The complex tasks in PBL often demand analytical and evaluative skills (Webb's DOK levels 3 and 4) as students engage with the problem, devise a solution, and reflect on their work.

The effectiveness of PBL is supported by psychological theories like the information processing theory, which highlights the role of active engagement in enhancing memory and recall. A study by Strobel and Van Barneveld (2009) found that PBL students show improved retention of knowledge, possibly due to the deep cognitive processing involved.

As cognitive scientist Daniel Willingham aptly puts it, "Memory is the residue of thought." PBL encourages learners to think critically and deeply, enhancing both learning and retention.

Here's a quick overview:

John Dewey : Emphasized learning through experience and the importance of problem-solving.
Maria Montessori : Advocated for child-centered, self-directed learning.
Jerome Bruner : Underlined learning as a social process and proposed the spiral curriculum.
Webb’s DOK : Supports PBL's encouragement of higher-order thinking skills.
Information Processing Theory : Reinforces the notion that active engagement in PBL enhances memory and recall.

This deep-rooted philosophical and psychological framework strengthens the validity of the problem-based learning approach, confirming its beneficial role in promoting valuable cognitive skills and fostering positive student learning outcomes.

What are the characteristics of problem-based learning?

Adding a little creativity can change a topic into a problem-based learning activity. The following are some of the characteristics of a good PBL model:

The problem encourages students to search for a deeper understanding of content knowledge;
Students are responsible for their learning. PBL has a student-centred learning approach . Students' motivation increases when responsibility for the process and solution to the problem rests with the learner;
The problem motivates pupils to gain desirable learning skills and to defend well-informed decisions ;
The problem connects the content learning goals with the previous knowledge. PBL allows students to access, integrate and study information from multiple disciplines that might relate to understanding and resolving a specific problem—just as persons in the real world recollect and use the application of knowledge that they have gained from diverse sources in their life.
In a multistage project, the first stage of the problem must be engaging and open-ended to make students interested in the problem. In the real world, problems are poorly-structured. Research suggests that well-structured problems make students less invested and less motivated in the development of the solution. The problem simulations used in problem-based contextual learning are less structured to enable students to make a free inquiry.

In a group project, the problem must have some level of complexity that motivates students towards knowledge acquisition and to work together for finding the solution. PBL involves collaboration between learners. In professional life, most people will find themselves in employment where they would work productively and share information with others. PBL leads to the development of such essential skills . In a PBL session, the teacher would ask questions to make sure that knowledge has been shared between pupils;
At the end of each problem or PBL, self and peer assessments are performed. The main purpose of assessments is to sharpen a variety of metacognitive processing skills and to reinforce self-reflective learning.
Student assessments would evaluate student progress towards the objectives of problem-based learning. The learning goals of PBL are both process-based and knowledge-based. Students must be assessed on both these dimensions to ensure that they are prospering as intended from the PBL approach. Students must be able to identify and articulate what they understood and what they learned.

Why is Problem-based learning a significant skill?

Using Problem-Based Learning across a school promotes critical competence, inquiry , and knowledge application in social, behavioural and biological sciences. Practice-based learning holds a strong track record of successful learning outcomes in higher education settings such as graduates of Medical Schools.

Educational models using PBL can improve learning outcomes by teaching students how to implement theory into practice and build problem-solving skills. For example, within the field of health sciences education, PBL makes the learning process for nurses and medical students self-centred and promotes their teamwork and leadership skills. Within primary and secondary education settings, this model of teaching, with the right sort of collaborative tools , can advance the wider skills development valued in society.

At Structural Learning, we have been developing a self-assessment tool designed to monitor the progress of children. Utilising these types of teaching theories curriculum wide can help a school develop the learning behaviours our students will need in the workplace.

Curriculum wide collaborative tools include Writers Block and the Universal Thinking Framework . Along with graphic organisers, these tools enable children to collaborate and entertain different perspectives that they might not otherwise see. Putting learning in action by using the block building methodology enables children to reach their learning goals by experimenting and iterating.

Scaffolding problem based learning with classroom tools

How is problem-based learning different from inquiry-based learning?

The major difference between inquiry-based learning and PBL relates to the role of the teacher . In the case of inquiry-based learning, the teacher is both a provider of classroom knowledge and a facilitator of student learning (expecting/encouraging higher-order thinking). On the other hand, PBL is a deep learning approach, in which the teacher is the supporter of the learning process and expects students to have clear thinking, but the teacher is not the provider of classroom knowledge about the problem—the responsibility of providing information belongs to the learners themselves.

As well as being used systematically in medical education, this approach has significant implications for integrating learning skills into mainstream classrooms .

Using a critical thinking disposition inventory, schools can monitor the wider progress of their students as they apply their learning skills across the traditional curriculum. Authentic problems call students to apply their critical thinking abilities in new and purposeful ways. As students explain their ideas to one another, they develop communication skills that might not otherwise be nurtured.

Depending on the curriculum being delivered by a school, there may well be an emphasis on building critical thinking abilities in the classroom. Within the International Baccalaureate programs, these life-long skills are often cited in the IB learner profile . Critical thinking dispositions are highly valued in the workplace and this pedagogical approach can be used to harness these essential 21st-century skills.

What are the Benefits of Problem-Based Learning?

Student-led Problem-Based Learning is one of the most useful ways to make students drivers of their learning experience. It makes students creative, innovative, logical and open-minded. The educational practice of Problem-Based Learning also provides opportunities for self-directed and collaborative learning with others in an active learning and hands-on process. Below are the most significant benefits of problem-based learning processes:

Self-learning: As a self-directed learning method, problem-based learning encourages children to take responsibility and initiative for their learning processes . As children use creativity and research, they develop skills that will help them in their adulthood.
Engaging : Students don't just listen to the teacher, sit back and take notes. Problem-based learning processes encourages students to take part in learning activities, use learning resources , stay active , think outside the box and apply critical thinking skills to solve problems.
Teamwork : Most of the problem-based learning issues involve students collaborative learning to find a solution. The educational practice of PBL builds interpersonal skills, listening and communication skills and improves the skills of collaboration and compromise.
Intrinsic Rewards: In most problem-based learning projects, the reward is much bigger than good grades. Students gain the pride and satisfaction of finding an innovative solution, solving a riddle, or creating a tangible product.
Transferable Skills: The acquisition of knowledge through problem-based learning strategies don't just help learners in one class or a single subject area. Students can apply these skills to a plethora of subject matter as well as in real life.
Multiple Learning Opportunities : A PBL model offers an open-ended problem-based acquisition of knowledge, which presents a real-world problem and asks learners to come up with well-constructed responses. Students can use multiple sources such as they can access online resources, using their prior knowledge, and asking momentous questions to brainstorm and come up with solid learning outcomes. Unlike traditional approaches , there might be more than a single right way to do something, but this process motivates learners to explore potential solutions whilst staying active.

Solving authentic problems using problem based learning

Embracing problem-based learning

Problem-based learning can be seen as a deep learning approach and when implemented effectively as part of a broad and balanced curriculum , a successful teaching strategy in education. PBL has a solid epistemological and philosophical foundation and a strong track record of success in multiple areas of study. Learners must experience problem-based learning methods and engage in positive solution-finding activities. PBL models allow learners to gain knowledge through real-world problems, which offers more strength to their understanding and helps them find the connection between classroom learning and the real world at large.

As they solve problems, students can evolve as individuals and team-mates. One word of caution, not all classroom tasks will lend themselves to this learning theory. Take spellings , for example, this is usually delivered with low-stakes quizzing through a practice-based learning model. PBL allows students to apply their knowledge creatively but they need to have a certain level of background knowledge to do this, rote learning might still have its place after all.

Key Concepts and considerations for school leaders

1. Problem Based Learning (PBL)

Problem-based learning (PBL) is an educational method that involves active student participation in solving authentic problems. Students are given a task or question that they must answer using their prior knowledge and resources. They then collaborate with each other to come up with solutions to the problem. This collaborative effort leads to deeper learning than traditional lectures or classroom instruction .

Key question: Inside a traditional curriculum , what opportunities across subject areas do you immediately see?

2. Deep Learning

Deep learning is a term used to describe the ability to learn concepts deeply. For example, if you were asked to memorize a list of numbers, you would probably remember the first five numbers easily, but the last number would be difficult to recall. However, if you were taught to understand the concept behind the numbers, you would be able to remember the last number too.

Key question: How will you make sure that students use a full range of learning styles and learning skills ?

3. Epistemology

Epistemology is the branch of philosophy that deals with the nature of knowledge . It examines the conditions under which something counts as knowledge.

Key question: As well as focusing on critical thinking dispositions, what subject knowledge should the students understand?

4. Philosophy

Philosophy is the study of general truths about human life. Philosophers examine questions such as “What makes us happy?”, “How should we live our lives?”, and “Why does anything exist?”

Key question: Are there any opportunities for embracing philosophical enquiry into the project to develop critical thinking abilities ?

5. Curriculum

A curriculum is a set of courses designed to teach specific subjects. These courses may include mathematics , science, social studies, language arts, etc.

Key question: How will subject leaders ensure that the integrity of the curriculum is maintained?

6. Broad and Balanced Curriculum

Broad and balanced curricula are those that cover a wide range of topics. Some examples of these types of curriculums include AP Biology, AP Chemistry, AP English Language, AP Physics 1, AP Psychology , AP Spanish Literature, AP Statistics, AP US History, AP World History, IB Diploma Programme, IB Primary Years Program, IB Middle Years Program, IB Diploma Programme .

Key question: Are the teachers who have identified opportunities for a problem-based curriculum?

7. Successful Teaching Strategy

Successful teaching strategies involve effective communication techniques, clear objectives, and appropriate assessments. Teachers must ensure that their lessons are well-planned and organized. They must also provide opportunities for students to interact with one another and share information.

Key question: What pedagogical approaches and teaching strategies will you use?

8. Positive Solution Finding

Positive solution finding is a type of problem-solving where students actively seek out answers rather than passively accept what others tell them.

Key question: How will you ensure your problem-based curriculum is met with a positive mindset from students and teachers?

9. Real World Application

Real-world application refers to applying what students have learned in class to situations that occur in everyday life.

Key question: Within your local school community , are there any opportunities to apply knowledge and skills to real-life problems?

10. Creativity

Creativity is the ability to think of ideas that no one else has thought of yet. Creative thinking requires divergent thinking, which means thinking in different directions.

Key question: What teaching techniques will you use to enable children to generate their own ideas ?

11. Teamwork

Teamwork is the act of working together towards a common goal. Teams often consist of two or more people who work together to achieve a shared objective.

Key question: What opportunities are there to engage students in dialogic teaching methods where they talk their way through the problem?

12. Knowledge Transfer

Knowledge transfer occurs when teachers use their expertise to help students develop skills and abilities .

Key question: Can teachers be able to track the success of the project using improvement scores?

13. Active Learning

Active learning is any form of instruction that engages students in the learning process. Examples of active learning include group discussions, role-playing, debates, presentations, and simulations .

Key question: Will there be an emphasis on learning to learn and developing independent learning skills ?

14. Student Engagement

Student engagement is the degree to which students feel motivated to participate in academic activities.

Key question: Are there any tools available to monitor student engagement during the problem-based curriculum ?

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Classroom Practice

Principles for Teaching Problem Solving

January 1998
Publisher: PLATO Learning, Inc.

Walden University

Indiana University Bloomington

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6 Tips for Teaching Math Problem-Solving Skills

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

Photo of elementary school teacher with students

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

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

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

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

1. Link problem-solving to reading

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

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

2. Avoid boxing students into choosing a specific operation

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

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

3. Revisit ‘representation’

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

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

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

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

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

4. Give time to process

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

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

5. Ask questions that let Students do the thinking

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

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

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

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

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

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

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

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

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

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

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

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

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

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

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

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

Key features of a good mathematics problem includes:

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

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

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

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

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

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

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

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

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

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

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

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

Number Talks

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

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

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

Inside Mathematics Number Talks
Number Talks Build Numerical Reasoning

Saying “This is Easy”

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

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

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

Instead, you and your students could say the following:

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

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

Using “Worksheets”

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

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

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

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

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

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

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

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

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

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

when we teach students how to problem solve

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

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

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

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Creative problem solving tools and skills for students and teachers

Creative Problem Solving: What Is It?

Creative Problem Solving, or CPS , refers to the use of imagination and innovation to find solutions to problems when formulaic or conventional processes have failed.

Despite its rather dry definition – creative problem-solving in its application can be a lot of fun for learners and teachers alike.

Why Are Creative Problem-Solving Skills Important?

By definition, creative problem-solving challenges students to think beyond the conventional and to avoid well-trodden, sterile paths of thinking.

Not only does this motivate student learning, encourage engagement, and inspire deeper learning, but the practical applications of this higher-level thinking skill are virtually inexhaustible.

For example, given the rapidly changing world of work, it is hard to conceive of a skill that will be more valuable than the ability to generate innovative solutions to the unique problems that will arise and that are impossible to predict ahead of time.

Outside the world of work, in our busy daily lives, the endless problems arising from day-to-day living can also be overcome by a creative problem-solving approach.

When students have developed their creative problem-solving abilities effectively, they will have added a powerful tool to attack problems that they will encounter, whether in school, work, or in their personal lives.

Due to its at times nebulous nature, teaching creative problem-solving in the classroom poses its own challenges. However, developing a culture of approaching problem-solving in a creative manner is possible.

In this article, we will take a look at a variety of strategies, tools, and activities that can help students improve their creative problem-solving skills.

The Underlying Principles of CPS

Before we take a look at a process for implementing creative problem solving, it is helpful to examine a few of the underlying principles of CPS. These core principles should be encouraged in the classroom. They are:

● Assume Nothing

Assumptions are the enemy of creativity and original thinking. If students assume they already have the answer, they will not be creative in their approach to solving a problem.

● Problems Are Opportunities

Rather than seeing problems as difficulties to endure, a shift in perspective can instead view problems as challenges that offer new opportunities. Encourage your students to shift their perspectives to see opportunities where they once saw problems.

● Suspend Judgment

Making immediate judgments closes down the creative response and the formation of new ideas. There is a time to make judgments, but making a judgment too early in the process can be very detrimental to finding a creative solution.

Cognitive Approaches: Convergent vs Divergent Thinking

“It is easier to tame a wild idea than it is to push a closer-in idea further out.”

— Alex Osborn

The terms divergent and convergent thinking, coined by psychologist J.P. Guilford in 1956, refer to two contrasting cognitive approaches to problem-solving.

Convergent Thinking can be thought of as linear and systematic in its approach. It attempts to find a solution to a problem by narrowing down multiple ideas into a single solution. If convergent thinking can be thought of as asking a single question, that question would be ‘ Why ?’

Divergent Thinking focuses more on the generation of multiple ideas and on the connections between those ideas. It sees problems as design opportunities and encourages the use of resources and materials in original ways. Divergent thinking encourages the taking of creative risks and is flexible rather than analytical in its approach. If it was a single question, it’d be ‘ Why not ?’

While it may appear that these two modes of thinking about a problem have an essentially competitive relationship, in CPS they can work together in a complementary manner.

When students have a problem to solve and they’re looking for innovative solutions, they can employ divergent thinking initially to generate multiple ideas, then convergent thinking to analyze and narrow down those ideas.

Students can repeat this process to continue to filter and refine their ideas and perspectives until they arrive at an innovative and satisfactory solution to the initial problem.

Let’s now take a closer look at the creative problem-solving process.

The Creative Problem-Solving Process

CPS helps students arrive at innovative and novel solutions to the problems that arise in life. Having a process to follow helps to keep students focused and to reach a point where action can be taken to implement creative ideas.

Originally developed by Alex Osborn and Sid Parnes, the CPS process has gone through a number of revisions over the last 50 or so years and, as a result, there are a number of variations of this model in existence.

The version described below is one of the more recent models and is well-suited to the classroom environment.

However, things can sometimes get a little complex for some of the younger students. So, in this case, it may be beneficial to teach the individual parts of the process in isolation first.

1. Clarify:

Before beginning to seek creative solutions to a problem, it is important to clarify the exact nature of that problem. To do this, students should do the following three things:

i. Identify the Problem

The first step in bringing creativity to problem-solving is to identify the problem, challenge, opportunity, or goal and clearly define it.

ii. Gather Data

Gather data and research information and background to ensure a clear understanding.

iii. Formulate Questions

Enhance awareness of the nature of the problem by creating questions that invite solutions.

Explore new ideas to answer the questions raised. It’s time to get creative here. The more ideas generated, the greater the chance of producing a novel and useful idea. At this stage in particular, students should be engaged in divergent thinking as described above.

The focus here shifts from ideas to solutions. Once multiple ideas have been generated, convergent thinking can be used to narrow these down to the most suitable solution. The best idea should be closely analyzed in all its aspects and further ideas generated to make subsequent improvements. This is the stage to refine the initial idea and make it into a really workable solution.

4. Implement

Create a plan to implement the chosen solution. Students need to identify the required resources for the successful implementation of the solution. They need to plan for the actions that need to be taken, when they need to be taken, and who needs to take them.

Summary of Creative Problem Solving Process

In each stage of the CPS Process, students should be encouraged to employ divergent and convergent thinking in turn. Divergent thinking should be used to generate multiple ideas with convergent thinking then used to narrow these ideas down to the most feasible options. We will discuss how students go about this, but let’s first take a quick look at the role of a group facilitator.

The Importance of Group Facilitator

CPS is best undertaken in groups and, for larger and more complex projects, it’s even more effective when a facilitator can be appointed for the group.

The facilitator performs a number of useful purposes and helps the group to:

Stay focused on the task at hand
Move through the various stages efficiently
Select appropriate tools and strategies

A good facilitator does not generate ideas themselves but instead keeps the group focused on each step of the process.

Facilitators should be objective and possess a good understanding of the process outlined above, as well as the other tools and strategies that we will look at below.

The Creative Problem-Solving Process: Tools and Strategies

There are several activities available to help students move through each stage. These will help students to stay on track, remove barriers and blocks, be creative, and reach a consensus as they progress through the CPS process.

The following tools and strategies can help provide groups with some structure and can be applied at various stages of the problem-solving process. For convenience, they have been categorized according to whether they make demands on divergent or convergent thinking as discussed earlier.

Divergent Thinking Tools:

● Brainstorming

Defined by Alex Osborn as “a group’s attempt to find a solution for a specific problem by amassing ideas ”, this is perhaps the best-known tool in the arsenal of the creative problem solver.

To promote a creative collaboration in a group setting, simply share the challenge with everyone and challenge them to come up with as many ideas as possible. Ideas should be concise and specific. For this reason, it may be worth setting a word limit for recording each idea e.g. express in headline form in no more than 5 words. Post-it notes are perfect for this.

You may also set a quota on the number of ideas to generate or introduce a time limit to further encourage focus. When completed, members of the group can share and compare all the ideas in search of the most suitable.

● 5 W’s and an H

The 5 W’s and an H are Who , What , Where , Why , and How . This strategy is useful to effectively gather data. Students brainstorm questions to ask that begin with each of the question words above in turn. They then seek to gather the necessary information to answer these questions through research and discussion.

● Reverse Assumptions

This activity is a great way to explore new ideas. Have the students begin by generating a list of up to 10 basic assumptions about the idea or concept. For each of these, students then explore the reverse of the assumption listing new insights and perspectives in the process.

The students can then use these insights and perspectives to generate fresh ideas. For example, an assumption about the concept of a restaurant might be that the food is cooked for you. The reverse of that assumption could be a restaurant where you cook the food yourself. So, how about a restaurant where patrons select their own recipes and cook their own food aided by a trained chef?

Convergent Thinking Tools

● How-How Diagram

This is the perfect activity to use when figuring out the steps required to implement a solution.

Students write the solution on the left-hand side of a page turned landscape. Working together, they identify the individual steps required to achieve this solution and write these to the right of the solution.

When they have written these steps, they go through each step one-by-one identifying in detail each stage of achieving that step. These are written branching to the right of each step.

Students repeat this process until they have exhausted the process and ended up with a comprehensive branch diagram detailing each step necessary for the implementation of the solution.

● The Evaluation Matrix

Making an evaluation matrix creates a systematic way of analyzing and comparing multiple solutions. It allows for a group to evaluate options against various criteria to help build consensus.

An evaluation matrix begins with the listing of criteria to evaluate potential solutions against. These can then be turned into the form of a positive question that allows for a Yes or No answer. For example, if the budget is the criteria, the evaluation question could be ‘ Is it within budget? ’

Make a matrix grid with a separate column for each of the key criteria. Write the positive question form of these criteria as headings for these columns. The different options can then be detailed and listed down the left-most column.

Students then work through each of the criteria for each option and record whether it fulfills, or doesn’t fulfill, each criteria. For more complex solutions, students could record their responses to each of the criteria on a scale from 0 to 5.

For example:

Using the example matrix above, it becomes very clear that Option 1 is the superior solution given that it completely fulfills all the criteria, whereas Option 2 and Option 3 fulfill only 2 out of the 3 criteria each.

● Pair & Share

This activity is suitable to help develop promising ideas. After making a list of possible solutions or questions to pursue, each individual student writes down their top 3 ideas.

Once each student has their list of their 3 best ideas, organize students into pairs. In their pairs, students discuss their combined 6 ideas to decide on the top 3 out of the 6. Once they have agreed on these, they write the new top 3 ideas on a piece of paper.

Now, direct the pairs of students to join up with another pair to make groups of 4. In these groups of 4, students discuss their collective 6 ideas to come up with a new list of the top 3 ideas.

Repeat this process until the whole class comes together as one big group to agree on the top 3 ideas overall.

Establish a Culture of Creative Problem Solving in the Classroom

Approaching problems creatively is about establishing a classroom culture that welcomes innovation and the trial and error that innovation demands. Too often our students are so focused on finding the ‘right‘ answer that they miss opportunities to explore new ideas.

It is up to us as teachers to help create a classroom culture that encourages experimentation and creative playfulness.

To do this we need to ensure our students understand the benefits of a creative approach to problem-solving.

We must ensure too that they are aware of the personal, social, and organizational benefits of CPS.

CPS should become an integral part of their approach to solving problems whether at school, work, or in their personal lives.

As teachers, it is up to us to help create a classroom culture that encourages experimentation and creative playfulness.

To do this, we must ensure our students understand the benefits of a creative approach to problem-solving.

CPS should become an integral part of their approach to solving problems, whether at school, work or in their personal lives.

Empowering Tomorrow’s Leaders: The Crucial Role of Computational and Systems Thinking in Education

the importance of systems thinking and computational thinking strategies for students cannot be overstated, as these skills are integral to navigating the complexities of our rapidly evolving digital landscape. Computational thinking, characterized by algorithmic problem-solving and logical reasoning, equips students with the ability to approach challenges systematically. In an era dominated by technology, these skills are not limited to coding but extend to critical thinking, enabling students to dissect problems, identify patterns, and devise efficient solutions. As our world becomes increasingly interconnected and data-driven, computational thinking provides a foundational framework for students to make sense of information, fostering a generation adept at leveraging technology for innovation.

Simultaneously, systems thinking is indispensable in comprehending the intricate web of relationships within various contexts. It encourages students to view issues holistically, understanding the interdependence of components and the ripple effects of decisions. In an era marked by global challenges, such as climate change and socio-economic disparities, systems thinking instills a proactive mindset. Students equipped with these skills are better prepared to analyze multifaceted problems, appreciate diverse perspectives, and collaborate on sustainable solutions.

Together, computational and systems thinking empower students to navigate an ever-changing world with confidence, adaptability, and a profound understanding of the interconnected systems that shape our future. These skills are not just academic; they are the building blocks of a resilient, innovative, and forward-thinking society.

be sure to check out our great video guides to teaching systems thinking and computational thinking below.

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Teaching Problem Solving in Math

Freebies , Math , Planning

$Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!$

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

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

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

The Problem Solving Strategies

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

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

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

There’s also this visuals strategy wheel practice.

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

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

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

The Problem Solving Steps

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

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

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

We did this over and over with example problems.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Act – Act on a strategy and try it out.

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

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

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

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

freebie , Math Workshop , Problem Solving

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Simple Guide to Problem-Solving Method of Teaching

You must be interested to know – What is the problem-solving method of teaching and how it works. We’ve explained its core principles, six-step process, and benefits with real-world examples.

Understand the Problem-Solving Method of Teaching

The basis of this modern teaching approach is to provide students with opportunities to face real-time challenges. It aims to help them understand how the concept behind a solution works in reality.

What is the Problem-Solving Method of Teaching?

The problem-solving method of teaching is a student-centered approach to learning that focuses on developing students’ problem-solving skills. In this method, students have to face real-world problems to solve.

They are encouraged to use their knowledge and skills to provide solutions. The teacher acts as a facilitator, providing guidance and support as needed, but ultimately the students are responsible for finding their solutions.

Must Read: How to Tell Me About Yourself in an Interview

5 Most Important Benefits of Problem-Solving Method of Teaching

The new way of teaching primarily helps students develop critical thinking skills and real-world application abilities. It also promotes independence and self-confidence in problem-solving.

The problem-solving method of teaching has several benefits. It helps students to:

#1 Enhances critical thinking

By presenting students with real-world problems to solve, the problem-solving method of teaching forces them:

– To think critically about the situation, and – To come up with their solutions.

This process helps students develop critical thinking skills essential for success in school and life.

#2 Fosters creativity

The problem-solving method of teaching encourages students to be creative in their problem-solving approach. There is often no one right answer to a problem, so students are free to come up with their unique solutions. This process helps students think creatively, an important skill in all areas of life.

#3 Encourages real-world application

The problem-solving method of teaching helps students learn how to apply their knowledge to real-world situations. By solving real-world problems, students can see:

– How their knowledge is relevant to their lives, – And, the world around them.

This helps students to become more motivated and engaged learners.

#4 Builds student confidence

When students can successfully solve problems, they gain confidence in their abilities. This confidence is essential for success in all areas of life, both academic and personal.

#5 Promotes collaborative learning

The problem-solving method of teaching often involves students working together to solve problems. This collaborative learning process helps students to develop their teamwork skills and to learn from each other.

Know 6 Steps in the Problem-Solving Method of Teaching

Also Read: Do You Know the Difference Between ChatGPT and GPT-4?

The problem-solving method of teaching typically involves the following steps:

Step 1: Identifying the problem

The first step is problem identification which students will be working on. This requires students to do the following:

– By presenting students with a real-world problem, or – By asking them to come up with their problems.

Step 2: Understanding the problem

Once students have identified the problem, they need to understand it fully. This may involve:

– Breaking the problem down into smaller parts, or – Gathering more information about the problem.

Step 3: Generating solutions

Once students understand the problem, they need to generate possible solutions. They have to do either of the following:

– By brainstorming, or – By exercising problem-solving techniques such as root cause analysis or the decision matrix.

Step 4: Evaluating solutions

Students need to evaluate the pros and cons of each solution before choosing one to implement.

Step 5: Implementing the solution

Once students have chosen a solution, they need to implement it. This may involve taking action or developing a plan.

Step 6: Evaluating the results

Once students have implemented the solution, they must evaluate the results to see if it was successful.

If the solution fails the expectations, students should re-run step 3 and generate new solutions.

Find Out Examples of the Problem-Solving Method of Teaching

Here are a few examples of how the problem-solving method of teaching applies to different subjects:

Math: Students face real-world problems such as budgeting for a family or designing a new product. Students would then need to use their math skills to solve the problem.
Science: Students perform a science experiment or research on a scientific topic to invent a solution to the problem. Students should then use their science knowledge and skills to solve the problem.
Social studies: Students analyze a historical event or current social issue and devise a solution. After that, students should exercise their social studies knowledge and skills to solve the problem.

How to Use Problem-Solving Methods of Teaching

Here are a few tips for using the problem-solving method of teaching effectively:

Choose problems that are relevant to students’ lives and interests.
Select those problems that are challenging but achievable.
Provide students with ample resources such as books, websites, or experts to solve the problem.
Motivate them to work collaboratively and to share their ideas.
Be patient and supportive. Problem-solving can be a challenging process, but it is also a rewarding one.

Also Try: 1-10 Random Number Generator

How to Choose: Let’s Draw a Comparison

The following table compares the different problem-solving methods:

Description	Pros	Cons
The teacher presents information to students who then complete exercises or assignments to practice the information.	– Simple and easy-to-follow	– Can be passive and boring for students
Students are presented with real-world problems to solve. They are encouraged to use their knowledge and skills to deliver solutions.	– Promotes active learning	– Can be challenging for students
Students are asked to investigate questions or problems. They are encouraged to gather evidence and come up with their conclusions.	– Encourages critical thinking	– Can be time-consuming

Which Method is the Most Suitable?

The most suitable way of teaching will depend on many factors such as the following:

– Subject matter, – Student’s age and ability level, and – Teacher’s preferences.

However, the problem-solving method of teaching is a valuable approach. It can be used in any subject area and with students of all ages.

Here are some additional tips for using the problem-solving method of teaching effectively:

Differentiate instruction. Not all students learn at the same pace or in the same way. Teachers can differentiate instruction to meet the needs of all learners by providing different levels of support and scaffolding.
Use formative assessment. Formative assessment helps track students’ progress and identify areas where they need additional support. Teachers can then use this information to provide students with targeted instruction.
Create a positive learning environment. Students need to feel safe and supported to learn effectively. Teachers can create a positive learning environment by providing students with opportunities for collaboration. They can celebrate their successes and create a classroom culture where mistakes are seen as learning opportunities.

Interested in New Tech: 7 IoT Trends to Watch in 2023

Some Unique Examples to Refer to Before We Conclude

Here are a few unique examples of how you incorporate the problem-solving method of teaching with different subjects:

English: Students analyze a grammar problem, such as a poem or a short story, and share their interpretation.
Art: Students can get a task to design a new product or to create a piece of art that addresses a social issue.
Music: Students write a song about a current event or create a new piece of music reflecting their cultural heritage.

Before You Leave

The problem-solving method of teaching is a powerful tool that can help students develop the skills they need to succeed in school and life. By creating a learning environment where students are encouraged to think critically and solve problems, teachers can help students to become lifelong learners.

Lastly, our site needs your support to remain free. Share this post on social media ( Linkedin / Twitter ) if you gained some knowledge from this tutorial.

Enjoy learning, TechBeamers.

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Data analytics: solving real-world problems one formula at a time with computational social science

photo illustration of computational science

The field of computational social science (CSS) unites researchers from multiple disciplines to analyze data from social media, historical archives, and other online sources to help solve critical global issues.

Unlike the data traditionally collected by social science researchers, this digital data often contains detailed information that can provide real-time insights into human behavior and social relationships.

This comprehensive feature explains what computational social science is, provides an overview of key domains in the field, and explores four practical applications of CSS that could help improve lives worldwide.

What is computational social science?

Computational social science represents an intersection of computer science and the social sciences, including political science, economics, sociology, psychology, and anthropology.

Technological advances have resulted in a massive volume of digital data continuously generated by online apps and social media platforms. Social science researchers trained in data analysis and programming can use this information to address complex problems such as socioeconomic inequalities, political division, and crime.

The computational tools and methods used in CSS include:

Data mining
Data visualization
Web scraping
Text analysis
Machine learning
Natural language processing (NLP)

Domains of computational social science

Without a strong background in computer science, it can be difficult to navigate the different facets and themes in the field of computational social science. Below are summaries of some of the key spheres of CSS.

Computational economics

In computational economics, computer scientists and economists use software and other computational tools to solve analytical problems and make more accurate predictions about the economy and financial markets.

Examples of the data analysis methods employed in computational economics include agent-based computational modeling and computational modeling of dynamic macroeconomic systems.

Social network analysis

Social network analysis is a theoretical approach and set of methods investigating complex social systems based on relationships, structures, location, roles, connectivity, matrix representations, and other network properties.

Key concepts in social network analysis include:

Homophily: The tendency for people to seek out those similar to themselves
Social contagion: The rapid spread of attitudes, emotions, and behaviors throughout groups without reason
Social capital : The theory that interpersonal relationships can create value

Cliodynamics

Cliodynamics is an interdisciplinary research area that integrates historical macro sociology, economic history, cultural and social evolution, long-term mathematical modeling, and the construction and analysis of historical databases to make predictions regarding issues such as social and political unrest, war, and violence.

The goal is to apply analytical, predictive scientific methods to social sciences (especially history) to identify patterns in social trends and forces that have continuously shaped society. However, the field is viewed with skepticism by many academic historians.

Culturomics

In the sub-field of culturomics, computational social science researchers study human behaviors, linguistic phenomena, and cultural trends by analyzing vast amounts of textual data from Google Books or other digital sources.

For instance, research that analyzed unique common words in the English lexicon in 1900, 1950, and 2000 found that in the latter year, more unique words were used in text than ever before. While studying the nature of celebrity over time, researchers found that people become more famous at an earlier age but are also forgotten at a faster rate.

CSS analytics: the role of X and digital media

Whenever users visit social media platforms such as X or other online media, they leave a digital footprint, including text, images, interaction metrics, and metadata that can be analyzed to make predictions about human social behavior. Researchers can combine these massive data sets with randomized survey responses that help them better understand human culture, including topics like how information spreads across networks.

For example, research on the viral spread of social media messages about autism spectrum disorders found that exposure to emotional language makes users more likely to become emotional themselves and identify with emotional content. Additionally, the use of angry language can contribute to the viral dissemination of misinformation.

Applying CSS and data analysis to 4 real-world problems

As mentioned above, melding data science techniques with social science research can lead to a much greater understanding of human social phenomena to help address real-world issues. Below are four such ways that computational social science tools are being used.

1. Supporting public health research

The goal of the field of public health is to improve the health of individuals and their communities. Public health professionals educate people on how to adopt healthy lifestyles, administer services, conduct disease research, and recommend policies to governments.

Computational social science research has many exciting applications to help bolster public health efforts. For example, members of the CSS Lab at the University of Pennsylvania used mobility data related to COVID-19 to conduct research and train epidemiological models to predict the impact of policies related to vaccination and reopening efforts during the pandemic.

More broadly, the Annual Sociology Review cites research by demographers who used digital data from Google Search and Facebook to analyze public health issues, including the prevalence of selective abortion in India, rates of binge drinking at US colleges, lifestyle disease in 47 countries, and patterns in fertility, mortality, and migration.

Computational health science is an emerging sub-field of CSS that harnesses advanced machine learning and graphical network-based analytics to provide insights into biological processes, clinical decision-making support, and the discovery of novel drugs and treatments. These efforts could have a significant impact on future public health outcomes.

2. Reducing political polarization and radicalization

There’s no question that political unrest and division is a major problem in the United States and abroad. According to an article in Greater Good Magazine , the many adverse effects of political polarization include:

Increased community segregation by political party
A more antagonistic political culture
A loss of trust in key institutions
An inability to come to a consensus on critical issues such as gun control
A rise in hate crimes and intergroup violence

In recent years, numerous scientific studies have employed computer science methods to analyze large data sets related to political science. Examples detailed in the Annual Sociology Review include:

Using Google Search data to study the demographic factors that can increase susceptibility to violent radicalization
Examining how elites in journalism, politics, and entertainment help shape global political discourse and drive the polarization of views
Analyzing data from social media and other platforms to better understand how digital tools have played a role in protests and other collective political action
Using online data, agent-based modeling, automated text analysis, and network-based research to study political tribalism and extremism, homophily, and the formation of “echo chambers” that limit exposure to opposing ideas
Performing experiments to learn how partisans can change their beliefs. Some findings include that exposure to opposing political views can increase partisanship while using specific moral language and matching linguistic styles may help reduce it.

3. Understanding perceptions around climate change

A 2023 report from UN Climate Change found that intergovernmental climate action plans are not effective enough to limit the rise of global temperatures to 1.5 degrees Celsius or meet the goals of the Paris Agreement. Greenhouse gas emissions need to be cut by 43 percent by 2030 to prevent dire impacts such as more frequent and harsh heatwaves, droughts, and rainfall.

According to Pew Research , perceptions in the US are polarized between the two major political parties, with nearly 78 percent of Democrats viewing climate change as a major threat compared to just 23 percent of Republicans. Climate change also falls in the top half of priority issues for Democrats and ranks second to last for Republicans.

A study using large-scale computational data and methods explored how the polarization of climate change politics in the US is influenced by a network of political and financial groups that actively work against climate change initiatives. The research found organizations with corporate funding were more likely to have created and amplified texts designed to polarize climate change issues, ultimately influencing public discourse.

In 2019 and 2020, Australian bushfires caused 33 human deaths, the displacement or death of close to 3 billion animals, and a devastating loss of habitat. This destruction attracted global public attention, and computational social science researchers sought to better understand the connection between the bushfires and public perceptions surrounding climate change.

Researchers used a data set of 9,000 tweets to identify and track keywords and hashtags to analyze perceptions of causality, blame, urgency, and prevention tactics related to the bushfires. Their results provided valuable insights; despite the spread of some forms of disinformation, X activity seemed to strengthen support for climate change action.

4. Improving insights into global poverty

According to the United Nations , the global poverty rate is expected to reach 7 percent by 2030, and many of the world’s vulnerable populations in low-income countries are not covered by social protections. Additionally, violent conflicts and the increasing impacts of climate change make it more difficult to combat poverty around the world.

Although industrialized nations have a plethora of demographic big data, one of the critical challenges to fighting poverty in the developing world is the lack of national statistical systems to identify location and socioeconomic status.

To help address this lack of data, CSS methods have been used in the following ways:

Scientists studied metadata from call logs of residential mobile phone users in Rwanda, analyzed their locations, and implemented a randomized survey to create a model to estimate poverty levels of households, regions, or entire nations
Researchers analyzed spatial data from Open Street Maps and the European Space Agency’s land use cover with five machine learning algorithms to predict poverty levels in Medellín, Colombia, which provides policymakers with a tool to make poverty and social interventions
Scholars confirmed that wealth and poverty estimates derived from three machine learning techniques outperformed those using the traditional linear mixed model

Models compiled by computational tools can be used by legislators and nonprofit organizations to better target humanitarian aid efforts in areas experiencing higher levels of poverty and use them to help assess other social inequalities in gender, race, ethnicity, health, and education.

Ready to enhance your digital literacy?

The Data Analytics and Psychological Sciences concentration for the Bachelor of Applied Arts and Sciences (BAAS) at Penn LPS Online explores the intersection of data analysis and the science of well-being.

In this liberal arts program, you’ll learn how well-being is measured, what contributes to human flourishing, and how to apply analytical and statistical methods to interpret and communicate data.

This Ivy League program prepares you to:

Understand advanced predictive modeling and machine learning
Implement and analyze basic regression models
Design surveys and experiments, including A/B tests
Develop data analysis in R and statistical programming skills
Explore key research themes in positive psychology and their relevance
Investigate applications of positive psychology in business, healthcare, education, and the nonprofit sector

If you haven’t already, apply to Penn LPS Online today and enroll in the Data Analytics and Psychological Sciences concentration for the BAAS degree. You can also view our course guide to learn more about what’s available in any upcoming term.

Additional sources

https://www.sydney.edu.au/arts/news-and-events/news/2022/06/23/what-is-computational-social-science.html

https://www.capterra.com/glossary/computational-economics/

https://comp-econ.com/

https://catalog.gmu.edu/courses/css/

https://macss.uchicago.edu/programs-of-study/courses

https://escholarship.org/uc/irows_cliodynamics

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Framework for a research-based and interdisciplinary use of sensors in elementary teacher education.

1. Introduction

Students’ intentionality and agency in using sensors to acquire diverse data, based on learning the affordances of sensors [ 16 ];
Didactic strategies that include problem-based collaborative situated learning [ 23 , 24 ], with collaborative learning of context-specific knowledge; concreteness fading [ 25 ], which scaffolds processes from concrete actions and observations to abstract representations; and embodied strategies [ 21 ], with multisensory approaches, using sensors as extended human senses [ 15 ];
Students’ collaborative participation in decision-making based on data collected with sensors, enhancing emotional and social development in learning contexts [ 26 ];
Inquiry strategies that promote epistemic practices, i.e., practices that construct knowledge and are similar to the practices of scientists, such as describing, predicting, using sensors, interpreting, organizing information, relating and making evidence-based decisions, and creating local knowledge [ 17 ].

2. Roles of Electronic Sensors in Elementary Teacher Education

2.1. the portuguese model of elementary teacher education, 2.2. electronic sensors as mainstream “things”, 2.3. electronic sensors as tools in technological and scientific practices.

Asking questions
Developing and using models
Planning and carrying out investigations
Analyzing and interpreting data
Using mathematics and computational thinking
Constructing explanations
Engaging in arguments from evidence
Obtaining, evaluating, and communicating information.

2.4. Electronic Sensors as Epistemic Mediators

2.5. electronic sensors as didactic tools, 2.6. electronic sensors as interdisciplinary tools, 3. research design.

A pragmatic worldview was recognized and assumed since the characteristics of the main goal require an educational problem-centered approach, which demands real-world practices focused on the consequences of actions [ 58 ].
A qualitative research method was adopted to acknowledge the complexity of the goal, allow emerging questions and practices, and collect data in the participants’ settings [ 58 ].
The analysis of case studies was the research option to support the framework design, since this framework should be situated, rely on multiple sources of evidence, and data collection and analysis should be guided by prior development of theoretical propositions [ 59 ].

4.1. Case Studies in Basic Education Degree Courses

Didactic Sequence Phases	Context	Contents	Strategies	Role of Sensors	Learning Outcomes
Conceptual exploration: photosynthesis and respiration Exploration of the Activity Guide	Pre-service teachers’ classroom activities	Photosynthesis; Respiration; Indoor air quality: pollutants and safety values	Group work		Knowledge of photosynthesis and respiration processes Knowledge of indoor air quality: pollutants and safety values
Exploratory use of sensors	Pre-service teachers’ classroom and outdoor curricular activities	Carbon dioxide sensors’ affordances	Group work	Didactic tools	Knowledge of carbon dioxide sensors’ affordances Competencies in the use of sensors
Laboratorial use of sensors	Pre-service teachers’ laboratory and curricular activities	Carbon dioxide in air: variables and measurements Photosynthesis, respiration, and Gas Exchange Multiple representations of environmental information Statistical measures	Laboratory group work	Laboratory/Scientific tools Epistemic tools	Sense of measurement of carbon dioxide in air Competencies in the use of multiple representations and statistical measures in data organization, processing, and interpretation
Reports’ development	Pre-service teachers’ autonomous curricular activities	Communication of environmental information and knowledge	Group work	Epistemic tools	Conceptual and procedural knowledge of the influence of plants on indoor air quality Communication and synthesis competencies

4.2. Case Study in Teacher Education Master’s Degree Interdisciplinary Course

Didactic Sequence Phases	Context	Contents	Strategies	Role of Sensors	Learning Outcomes
Activity planning Exploratory use of heart rate sensor	Pre-service teachers’ classroom activities	Heart rate Heart rate in different contexts and activities Heart rate sensors’ affordances	Group work	Didactic tools	Knowledge of heart rate in different contexts and activities Knowledge of heart rate sensors’ affordances in the context of Mathematics and Natural Sciences collaborative activities Competencies in the use of sensors
Experimental activities: data collection, processing, and interpretation	Pre-service teachers’ classroom and outdoor curricular activities Teacher mediation	Heart rate and location variables and measurements Multiple representations of heart rate and location information Statistical measures Methods and techniques of data organization and processing	Fieldwork: data collection Use of Excel in data processing	Scientific tools Epistemic tools Interdisciplinary tools	Knowledge of heart rate in different contexts and activities Sense of number and measurement of heart rate and location Competencies in the use of multiple representations and statistical measures in data organization and processing Competencies in the use of spreadsheets in the context of health and environmental tasks.
Reports’ development	Pre-service teachers in autonomous work	Communication of health and environmental information and knowledge	Communication in the activity process	Epistemic tools	Communication and synthesis competencies

Didactic Sequence Phases	Context	Contents	Strategies	Role of Sensors	Learning Outcomes
Exploration of the Activity Guide Inquiry planning	Pre-service teachers’ classroom activities	Sound level Sound problems in schools	Group work		Knowledge of sound level and sound problems in schools Inquiry planning competencies
Interdisciplinary use of sensors: exploratory activities	Pre-service teachers’ classroom and outdoor curricular activities	Sound sensors’ affordances	Practical work: use of sensors	Didactic tools	Knowledge of sound sensors’ affordances in the context of Mathematics and Natural Sciences collaborative environmental activities Competencies in the use of sensors
Inquiry activities: planning, data collection, processing, and interpretation	Pre-service teachers’ classroom and outdoor curricular activities Teacher mediation	Sound level variables and measurements in school Multiple representations of environmental information Statistical measures Methods and techniques of data organization and processing	Fieldwork: data collection Use of Excel in data processing	Interdisciplinary tools Scientific tools Epistemic tools	Sense of number and measurement of sound level Competencies in the use of multiple representations and statistical measures in data organization and processing Competencies in the use of spreadsheets in the context of environmental inquiry tasks
Inquiry reports: presentations and debate	Pre-service teachers in classroom activities	Communication of environmental information and knowledge	Communication in the inquiry process	Epistemic tools	Communication and synthesis competencies

Didactic Sequence Phases	Context	Contents	Strategies	Role of Sensors	Learning Outcomes
Conceptual exploration: temperature, relative humidity, and thermal comfort	Fifth-grade children, mediated by pre-service teachers during classroom activities	Temperature and Relative Humidity Thermal comfort problems in schools	Debate in whole class Individual registration of concepts Sharing of concepts Systematization of concepts		Knowledge of temperature, relative humidity, and thermal comfort problems in schools
Interdisciplinary use of sensors Inquiry activities: data collection	Fifth-grade children, mediated by pre-service teachers in classroom and outdoor curricular activities	Temperature and relative humidity sensors’ affordances	Practical work: use of sensors in several locations, following clues (spatial orientation) Fieldwork: data collection	Didactic tools Interdisciplinary tools	Knowledge of temperature and relative humidity sensors’ affordances in the context of Mathematics and Natural Sciences collaborative environmental activities Competencies in the use of sensors
Inquiry activities: data processing and interpretation	Fifth-grade children, mediated by pre-service teachers in classroom activities	Temperature and relative humidity variables and measurements in school Percentages Multiple representations of environmental information Statistical measures Coordinates	Group work: data recording in a thermal comfort diagram	Scientific tools Epistemic tools	Sense of number and measurement of temperature and relative humidity Competencies in the use of multiple representations and statistical measures in data processing Competencies in the use of coordinates in a Cartesian system.
Inquiry reports: presentations and debate	Fifth-grade children, mediated by pre-service teachers in classroom activities	Communication of environmental information and knowledge	Communication in the inquiry process Comparison of the results obtained by the groups	Epistemic tools	Communication and synthesis competencies

4.3. Case Studies in Internships of the Master’s Degree in Teaching

Didactic Sequence Phases	Context	Contents	Strategies	Role of Sensors	Learning Outcomes
Conceptual exploration: Climate change and biodiversity	Internship: fifth-grade classroom activity	Climate change Biodiversity	Debate, brainstorming		Identification of students’ prior knowledge about climate change and biodiversity
Experimental activity: Concentration of carbon dioxide on different air samples	Internship: fifth-grade classroom activity	Carbon dioxide concentration in air Air quality	Laboratory group work	Didactic tools Laboratory/Scientific tools Epistemic tools	Knowledge of carbon dioxide in air sensors’ affordances Competencies in the use of sensors Sense of measurement of carbon dioxide in air
Experimental activity: Measurement of carbon dioxide in air and water pH in eco-chambers before and after the combustion of a candle	Internship: fifth-grade classroom activity	Water pH Combustion	Laboratory group work	Didactic tools Laboratory/Scientific tools Epistemic tools	Knowledge of pH and carbon dioxide in air sensors’ affordances Competencies in the use of sensors Sense of measurement of pH in water and carbon dioxide in air
Experimental activity: Measurement of water pH in eco-chambers after exhaled air	Internship: fifth-grade classroom activity	Water pH Breathing	Laboratory group work	Didactic tools Laboratory/Scientific tools Epistemic tools	Knowledge of the affordances of pH sensors and carbon dioxide in air Competencies in the use of sensors Sense of measurement of pH in water and carbon dioxide in air
Systematization of the whole process and learning outcomes	Internship: fifth-grade classroom activity	Climate change Biodiversity	Debate Questionnaire	Epistemic tools	Communication Synthesis competencies Knowledge acquisition about climate change and biodiversity

Didactic Sequence Phases	Context	Contents	Strategies	Role of Sensors	Learning Outcomes
Introduction to the temperature and relative humidity concepts. Thermal comfort multisensory exploration	Internship: third-grade classroom activity	Temperature and relative humidity	Group work		Basic knowledge and multisensory awareness of temperature and relative humidity
Data collection with sensors and interpretation and identification of problems in their school	Internship: third-grade outdoor and indoor activities	Thermal comfort, temperature, and relative humidity	Experimental work on thermal comfort in several locations	Didactic tools Scientific tools Epistemic tools	Knowledge of thermal comfort problems in school Knowledge of thermal comfort sensors’ affordances Competencies in the use of sensors Competencies in the use of multiple representations and statistical measures in data processing Competencies in the use of coordinates in a Cartesian system
Reflection on the problems found and possible improvement solutions	Internship: third-grade classroom activity	Thermal comfort, temperature, and relative humidity	Group work	Epistemic tools	Formulation of solutions to the identified thermal comfort problems in school
Sharing the improvements/solutions found	Internship: third-grade classroom activity	Thermal comfort, temperature, and relative humidity	Debate Communication	Epistemic tools	Communication and synthesis competencies Knowledge acquisition about thermal comfort

4.4. Case Studies in Teacher Education of In-Service Teachers

Didactic Sequence Phases	Context	Contents	Strategies	Role of Sensors	Learning Outcomes
Conceptual exploration: sound	First- and fourth-grade classroom activity Outdoor curricular activities	Sound	Debate in whole class Outdoor listening activity with distinction between pleasant and unpleasant sounds		Knowledge of sound and sound problems in schools
Formulation of predictions	First- and fourth-grade classroom activity	Sound level Sound problems in schools	Collective and individual predictions		Knowledge of sound level and sound problems in schools
Interdisciplinary use of sensors: exploratory activities Inquiry activities: data collection	Outdoor curricular activities	Sound level Sound sensors’ affordances	Group work Fieldwork: data collection	Didactic tools Interdisciplinary tools	Knowledge of sound level Competencies in the use of sensors
Inquiry activities: data processing and interpretation	First- and fourth-grade classroom activity, Teacher mediation	Sound level variables and measurements in school Multiple representations of environmental information Statistical measures	Data analysis Confrontation between the predictions and the measured values	Interdisciplinary tools Scientific tools Epistemic tools	Sense of number and measurement of sound level Competencies in the use and interpretation of multiple representations and statistical measures in data processing
Inquiry reports: presentations and debate Creation, dissemination, and implementation of solutions	First- and fourth-grade classroom activity, Teacher mediation	Communication of environmental information and knowledge Sound problem solutions in schools	Communication in the inquiry process Children’s agency in solutions and their implementation	Epistemic tools	Communication and synthesis competencies Competencies in citizen intervention

5. Discussion: The Roles of Environmental Sensors in Teacher Education

6. conclusions: framework for the use of sensors in the diverse areas of teacher education, author contributions, institutional review board statement, informed consent statement, data availability statement, conflicts of interest.

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Query Exercise: Solving The 201 Buckets Problem

When you run a query, SQL Server needs to estimate the number of matching rows it’ll find – so that it can decide which indexes to use, whether to go parallel, how much memory to grant, and more.

For example, take any Stack Overflow database , and let’s say I have an index on Location, and I want to find the top-ranking users in Lithuania:

INDEX Location ON dbo.Users(Location); * FROM dbo.Users Location = N'lithuania' BY Reputation DESC;

Then SQL Server has to guess how many people are in Lithuania so it can decide whether to use the index on Location, or do a table scan – because if there are a lot of folks in Lithuania, then it would mean a lot of key lookups to get the Reputation value for each of them.

We’ll run the query in the small StackOverflow2010 database and review the actual execution plan :

In the top right operator, the Index Seek, SQL Server only estimated 5 rows, but 84 rows actually came back. Now, that’s not really a problem for this particular query because:

SQL Server used the index – which makes the query fast
SQL Server did 84 key lookups instead of 5 – but still, that’s less logical reads than a table scan
The query went single-threaded – but there was so little work that it didn’t matter
The query didn’t spill to disk – there’s no yellow bang on the sort operator

As our database grows, though, the lines start to blur. Let’s run the same query on the largest current version of the StackOverflow database and see what happens in the actual execution plan :

The top right operator, the Index Seek, shows just 8 rows estimated, but 2,554 rows were actually found. As our data size grows, these estimate variances start to become problematic. Now granted, this succeeds in the same way the 2010 query succeeds: we get an index seek, it’s still less logical reads than a key lookup plan would be, the single-threaded thing isn’t a problem for a 27 millisecond query, and we don’t spill to disk.

However, if we start to join to other tables (and we will, in the next Query Exercise), then this under-estimation is going to become a problem.

Why is the estimate wrong?

We do indeed have statistics on the Location index, and they were created with fullscan since we just created the index. Let’s view the statistics for the large database:

SHOW_STATISTICS('dbo.Users', 'Location')

And check out the histogram contents – we’ll page down to Lithuania:

Or rather, we’ll page down to where you would expect Lithuania to be, and there’s a problem: Lithuania’s not there. SQL Server’s statistics are limited to just 201 buckets, max. (Technically, it’s up to 200 buckets for “normal” values in the table, plus 1 bucket for null.)

SQL Server does the best job it can of picking outliers in order to paint a perfect picture of the data, but it’s hard with just 201 buckets.

Typically – but not always – when SQL Server picks the locations that it’ll use for outliers, it uses around the top 200 locations by size, but this can vary a lot depending on the sort order of the column and the distribution of the data. Let’s look at the top locations:

TOP 250 Location, COUNT(*) AS recs FROM dbo.Users GROUP BY Location ORDER BY COUNT(*) DESC;

And Lithuania is at row 240 in this case:

So it’s a big location – but not big enough to hit the top 201 , which means it’s not going to get accurate estimates. The estimates are derived by looking at which bucket Lithuania is in – in the screenshot below, it’s row 100:

Lithuania is higher than Lisbon, but less than London, so it’s in the row 100 bucket. The row 100’s AVG_RANGE_ROWS is 7.847202, which means that any location between Lisbon and London has an average number of rows of about 8. And that’s where the estimate is coming from in our query:

Your challenge: get an accurate estimate.

You can change the query, the database, server-level settings, you name it. Anything that you would do in a real-life situation, you can do here. However, having done this exercise in my Mastering classes, I can tell you a couple things that people will try to do, but don’t really make sense.

You don’t wanna dump the data into a temp table first. Sometimes people will extract all of the data into a temp table, and then select data out of the temp table and say, “See, the estimate is accurate!” Sure it is, speedy, but look at your estimate from when you’re pulling the data out of the real table – the estimate’s still wrong there.

You don’t wanna use a hard-coded stat or index for just ‘Lithuania’. That only solves this one value, but you’ll still have the problem for every other outlier. We’re looking for a solution that we can use for most big outliers. (It’s always tricky to phrase question requirements in a way that rules out bad answers without pointing you to a specifically good answer, hahaha.)

Put your queries in a Github Gist and the query plans in PasteThePlan , showing your new accurate estimates , and include those link in your comments. Check out the solutions from other folks, and compare and contrast your work. I’ll circle back next week for a discussion on the answers. Have fun!

10 Comments . Leave new

My Gist https://gist.github.com/Paul-Fenton/83b0829263e9586868e1bd29fc2d6ccf

Query Plan https://www.brentozar.com/pastetheplan/?id=r19_spQjC

Create a new column “PopularLocation” which is set to the Location if it’s one of the top 250 locations.

Then change the query to look like:

SELECT * FROM dbo.Users WHERE Location = N’Lithuania’ OR PopularLocation = N’Lithuania’ ORDER BY Reputation DESC;

The estimate is now “84 of 85 rows (98%)” instead of “84 of 5 rows (1680%)”

I love the creativity! But…

Now the query is doing a table scan for just 84 rows. Check the estimates on other values, like India or San Diego

Sorry, cannot test it right now, just thinking out loud, perhaps filtered stats may solve it… I know it won’t probably work for other locations, but perhaps if we use parameter for the location, and using RECOMPILE hint that will be more accurate… Again sorry , but cannot test it right now…..

Uri, please reread the post more carefully and respect the time of others. Thank you.

Sometimes I need to cheat mssql because I want to see what will happen when there are many rows.

UPDATE STATISTICS dbo.Users([Location]) WITH rowcount =500000000000;

This gives 77 rows in Lithuanian. I known its not real life but for testing it is sometimes useful.

HAHAHA! That’s a funny idea, love it.

Code: https://gist.github.com/samot1/8d6e3bc5feee76b7959841cae8f81b97 Plan: https://www.brentozar.com/pastetheplan/?id=HJnkAt4oR

Idea: I just created a filtered statistic for every location starting with A, with B, with C … with Z

This way I have not just 200 statistic steps but 5200 which is enough to cover the most larger locations and get the correct estimates.

Drawback: you may or may not have to use OPTION(RECOMPILE) to use the correct statistic. When you put the location into a variable and use it in the query or have the PARAMETERIZATION option on your database set to FORCED you always have to use RECOMPILE to get the correct estimate.

PS: in a real scenario there may be much better options to archive the correct estimate, but you would need to know, how exactly the database is queried (which queries, how often, what happens with the results, how busy is the server …) and use all this information to decide the best option (which could be to simply ignore the wrong estimates too, since the overhead is bigger than the result)

Thomas – that is intriguing! I love the creativity.

I like the drawbacks that you put in the Gist – there are totally drawbacks here, for sure. For example, each stat has to be updated separately, so we just made our maintenance window explode. However, if someone is advanced enough to use this solution, I’d also expect them to be advanced enough to have relatively infrequent, targeted statistics updates.

I like it! I think you’ll also like the solution I talk about in next week’s post.

I had another much worse idea but it didn’t work (even if it could/should): CREATE PARTITION FUNCTION pf_a_to_z (NVARCHAR(100)) AS RANGE RIGHT FOR VALUES (‘A’, ‘B’, ‘C’, ‘D’, ‘E’, ‘F’, ‘G’, ‘H’, ‘I’, ‘J’, ‘K’, ‘L’, ‘M’, ‘N’, ‘O’, ‘P’, ‘Q’, ‘R’, ‘S’, ‘T’, ‘U’, ‘V’, ‘W’, ‘X’, ‘Y’, ‘Z’) CREATE PARTITION SCHEME ps_a_to_z AS PARTITION pf_a_to_z ALL TO ([PRIMARY]) ALTER TABLE [dbo].[Users] DROP CONSTRAINT [PK_Users_Id]; — drop old PK

CREATE UNIQUE CLUSTERED INDEX pk_Users_id ON dbo.Users (Id, Location) ON ps_a_to_z (Location) — base table (=clustered index) must be partition aligned, to allow STATISTICS_INCREMENTAL = ON on the location index GO CREATE NONCLUSTERED INDEX [Location] ON [dbo].[Users] ([Location]) WITH (DROP_EXISTING = ON, DATA_COMPRESSION = NONE, SORT_IN_TEMPDB = ON, ALLOW_ROW_LOCKS = ON, ALLOW_PAGE_LOCKS = ON, STATISTICS_INCREMENTAL = ON ) ON ps_a_to_z (Location); GO

now I have a table who is partitioned by the location, the STATISTICS_INCREMENTAL = ON says, that it should get one statistic per partition (plus a combined one), but sadly the SQL server eliminates the statistics all besides one but didn’t use the partitions statistic, so the estimates are still wrong.

And I have several drawbacks, e.g. I need to specifiy the location to be able to query the UserID efficent and I can have the same UserId twice in several locations, except I create another unique nonclustered index on the UserId …

[…] Query Exercise: Solving The 201 Buckets Problem (Brent Ozar) […]

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Explicit iterative algorithms for solving the split equality problems in Hilbert spaces

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Published: 20 August 2024

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Truong Minh Tuyen 1 &
Nguyen Song Ha 1

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We introduce and study some explicit iterative algorithms for solving the system of split equality problems in Hilbert spaces. The strong convergence of the proposed algorithms is proved by using some milder conditions put on control parameters than the one used in Tuyen (Bull Malays Math Sci Soc 46:44, 2023).

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On the convergence of CQ algorithm with variable steps for the split equality problem

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Acknowledgements

All the authors are grateful to the editors and to an anonymous referee for their useful comments and helpful suggestions.

Truong Minh Tuyen and Nguyen Song Ha were supported by the Science and Technology Fund of the Thai Nguyen University of Sciences.

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Teaching problem solving
Working on solutions. In the solution phase, one develops and then implements a coherent plan for solving the problem. As you help students with this phase, you might ask them to: identify the general model or procedure they have in mind for solving the problem. set sub-goals for solving the problem. identify necessary operations and steps.
Teaching problem solving: Let students get 'stuck' and 'unstuck'
Teaching problem solving: Let students get 'stuck' and 'unstuck'. This is the second in a six-part blog series on teaching 21st century skills, including problem solving , metacognition ...
The effectiveness of collaborative problem solving in promoting
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 ...
Critical Thinking and Problem-Solving
Critical thinking involves asking questions, defining a problem, examining evidence, analyzing assumptions and biases, avoiding emotional reasoning, avoiding oversimplification, considering other interpretations, and tolerating ambiguity. Dealing with ambiguity is also seen by Strohm & Baukus (1995) as an essential part of critical thinking ...
The process of implementing problem-based learning in a teacher
This is an example of students learning by solving problems using PBL and reflecting on their experiences (Barrows & Tamblyn ... reflective abilities, and teamwork were enhanced through the use of a PBL teaching approach and practical application of PBL. Pre-service teachers also developed a deeper understanding of the Principles of Instruction ...
Problem based learning: a teacher's guide
Problem-based learning (PBL) is a style of teaching that encourages students to become the drivers of their learning process. Problem-based learning involves complex learning issues from real-world problems and makes them the classroom's topic of discussion; encouraging students to understand concepts through problem-solving skills rather than ...
(PDF) Principles for Teaching Problem Solving
structured problem solving. 7) Use inductive teaching strategies to encourage synthesis of mental models and for. moderately and ill-structured problem solving. 8) Within a problem exercise, help ...
Problem Solving and Teaching How to Solve Problems in Technology-Rich
By drawing from the literature on technological pedagogical content knowledge, design thinking, general and specific methods of problem solving, and role of technologies for solving problems, this article highlights the importance of problem solving for future teachers and discusses strategies that can help them become good problem solvers and ...
PDF Problem Based Learning: A Student-Centered Approach
energies towards solving them. Although Problem-based learning has appeared since the dawn of time, in higher education.Now in the 21st century teachers and in other professionals across the globe using PBL in various disciplines. Many teachers in higher education are now highly experienced in the design and use of problems and
Elementary teachers' experience of engaging with Teaching Through
Teaching Through Problem Solving (TTP) is considered a powerful means of promoting mathematical understanding as a by-product of solving problems, where the teacher presents students with a specially designed problem that targets certain mathematics content (Stacey, 2018; Takahashi et al., 2013).The lesson implementation starts with the teacher presenting a problem and ensuring that students ...
6 Tips for Teaching Math Problem-Solving Skills
Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution. 1. Link problem-solving to reading. When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving ...
Full article: Understanding and explaining pedagogical problem solving
Purpose . This theoretical paper builds on the Pedagogy Analysis Framework by integrating it with pedagogical problem-solving theory, illustrating the resultant extended Pedagogy Analysis Framework and Pedagogical Problem Typology using data from a video-based study of one science and one Religious Education (RE) lesson.
Teaching Mathematics Through Problem Solving
Teaching about problem solving begins with suggested strategies to solve a problem. For example, "draw a picture," "make a table," etc. You may see posters in teachers' classrooms of the "Problem Solving Method" such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no ...
Creative problem solving tools and skills for students and teachers
So, in this case, it may be beneficial to teach the individual parts of the process in isolation first. 1. Clarify: Before beginning to seek creative solutions to a problem, it is important to clarify the exact nature of that problem. To do this, students should do the following three things: i. Identify the Problem.
Teaching the IDEAL Problem-Solving Method to Diverse Learners
Problem-solving is the capacity to identify and describe a problem and generate solutions to fix it. Problem-solving involves other executive functioning behaviors as well, including attentional control, planning, and task initiation. Individuals might use time management, emotional control, or organization skills to solve problems as well.
Teaching Problem Solving in Math
Step 1 - Understand the Problem. To help students understand the problem, I provided them with sample problems, and together we did five important things: read the problem carefully. restated the problem in our own words. crossed out unimportant information. circled any important information.
Problem-Solving Method of Teaching Made Easy
The problem-solving method of teaching is a student-centered approach to learning that focuses on developing students' problem-solving skills. In this method, students have to face real-world problems to solve. They are encouraged to use their knowledge and skills to provide solutions. The teacher acts as a facilitator, providing guidance and ...
5: Problem Solving
5.1: Problem Solving An introduction to problem-solving is the process of identifying a challenge or obstacle and finding an effective solution through a systematic approach. It involves critical thinking, analyzing the problem, devising a plan, implementing it, and reflecting on the outcome to ensure the problem is resolved.
Problem solving by students with intellectual disability.
This chapter concerns with cognitive engagement in students with intellectual disability (ID) when learning to read, and more specifically, with four observable components of cognitive engagement: Selection, elaboration, monitoring, and problem solving. The observable behaviours of these components include linking the presented stimuli to prior knowledge and requesting clarification ...
Quiz
Give adequate time to solve problems. Use worthwhile tasks. Model a similar problem. Question 7 2 / 2 pts Providing students with a list of area formulas and asking them to ±nd the area of a given rectangle. Having students develop their own word problems that use a recently learned algorithm.
Data analytics: solving real-world problems one formula at a time with
Additionally, the use of angry language can contribute to the viral dissemination of misinformation. Applying CSS and data analysis to 4 real-world problems. As mentioned above, melding data science techniques with social science research can lead to a much greater understanding of human social phenomena to help address real-world issues.
Framework for a Research-Based and Interdisciplinary Use of ...
In the present research, the use of sensors as scientific, epistemic, and interdisciplinary tools, as well as didactic tools, by pre-service and in-service teachers in scientific inquiry activities to solve real local environmental problems revealed potentialities in different areas of initial teacher education, namely Training in the teaching ...
Query Exercise: Solving The 201 Buckets Problem
When you run a query, SQL Server needs to estimate the number of matching rows it'll find - so that it can decide which indexes to use, whether to go parallel, how much memory to grant, and more. For example, take any Stack Overflow database, and let's say I have an index on Location, and...
Explicit iterative algorithms for solving the split equality problems
From now on, we consistently use the notation $\mathbb H:=H_1\times H_2$. Assume that all the conditions from (D1) to (D4) hold. We know that the SSEP is an ill-posed problem, and one of the popular methods for solving the class of the ill-posed problem is the Tikhonov regularization method.

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Problem-Solving Method in Teaching

Table of Contents

Meaning of Problem-Solving Method

Benefits of Problem-Solving Method

Steps in Problem-Solving Method

Verification of the concluded solution or Hypothesis

Why Every Educator Needs to Teach Problem-Solving Skills

Want to learn more about how to measure and teach students’ higher-order skills, including problem solving, critical thinking, and written communication?

Problem-Solving Skills Help Students…

Set and Achieve Goals

Resolve Conflicts

Achieve Success

Problem-Solving Skills Can Be Measured and Taught

What Are Performance-Based Assessments?

Preview a Performance-Based Assessment

Performance-Based Assessments Help Students Build and Practice Problem-Solving Skills

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Teaching Problem-Solving Skills

Principles for teaching problem solving

Woods’ problem-solving model

Think about it

Plan a solution

Carry out the plan

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Don’t Just Tell Students to Solve Problems. Teach Them How.

Published Date

Not getting stuck

What does it mean to explicitly teach problem solving?

Portrait of the Problem-Solving Curriculum

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Center for Teaching Innovation

Problem-Based Learning

Why Use Problem-Based Learning?

Considerations for Using Problem-Based Learning

Getting Started with Problem-Based Learning

Teaching problem solving

Introducing the problem

Working on solutions

Teaching problem solving: Let students get ‘stuck’ and ‘unstuck’

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

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