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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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.

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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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.

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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. 

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.

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

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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Problem solving is a critical skill for technical education and technical careers of all types. But what are best practices for teaching problem solving to high school and college students? 

The University of California San Diego Jacobs School of Engineering is on the forefront of efforts to improve how problem solving is taught. This UC San Diego approach puts hands-on problem-identification and problem-solving techniques front and center. Over 1,500 students across the San Diego region have already benefited over the last three years from this program. In the 2023-2024 academic year, approximately 1,000 upper-level high school students will be taking the problem solving course in four different school districts in the San Diego region. Based on the positive results with college students, as well as high school juniors and seniors in the San Diego region, the project is getting attention from educators across the state of California, and around the nation and the world.

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In Summer 2023, th e 27 community college students who took the unique problem-solving course developed at the UC San Diego Jacobs School of Engineering thrived, according to Alex Phan PhD, the Executive Director of Student Success at the UC San Diego Jacobs School of Engineering. Phan oversees the project. 

Over the course of three weeks, these students from Southwestern College and San Diego City College poured their enthusiasm into problem solving through hands-on team engineering challenges. The students brimmed with positive energy as they worked together. 

What was noticeably absent from this laboratory classroom: frustration.

“In school, we often tell students to brainstorm, but they don’t often know where to start. This curriculum gives students direct strategies for brainstorming, for identifying problems, for solving problems,” sai d Jennifer Ogo, a teacher from Kearny High School who taught the problem-solving course in summer 2023 at UC San Diego. Ogo was part of group of educators who took the course themselves last summer.

The curriculum has been created, refined and administered over the last three years through a collaboration between the UC San Diego Jacobs School of Engineering and the UC San Diego Division of Extended Studies. The project kicked off in 2020 with a generous gift from a local philanthropist.

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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Why Teaching Problem-Solving Skills is Essential for Student Success

Teaching the art of problem-solving is crucial for preparing students to thrive in an increasingly complex and interconnected world. Beyond the ability to find solutions, problem-solving fosters critical thinking, creativity, and resilience: qualities essential for academic success and lifelong learning. 

This article explores the importance of problem-solving skills, critical strategies for nurturing them in students, and practical approaches educators and parents can employ. 

By equipping students with these skills, we empower them to tackle challenges confidently, innovate effectively, and contribute meaningfully to their communities and future careers .

Why Teaching Problem-Solving Skills is Important

Problem-solving is a crucial skill that empowers students to tackle challenges with confidence and creativity . In an educational context, problem-solving is not just about finding solutions; it involves critical thinking, analysis, and application of knowledge. Students who excel in problem-solving can understand complex problems, break them down into manageable parts, and develop effective strategies to solve them. This skill is applicable across all subjects, from math and science to language arts and social studies, fostering a more profound understanding and retention of material .

Beyond academics, problem-solving is a cornerstone of success in life. Successful people across various fields possess strong problem-solving abilities. They can navigate obstacles, innovate solutions, and adapt to changing circumstances. In engineering and business management careers, problem solvers are highly valued for their ability to find efficient and creative solutions to complex issues.

Educators prepare students for future challenges and opportunities by teaching problem-solving in schools. They learn to think critically , work collaboratively, and persist in facing difficulties, all essential lifelong learning and achievement skills. Thus, nurturing problem-solving skills in students enhances their academic performance and equips them for success in their future careers and personal lives.

Aspects of Problem Solving

Developing problem-solving skills is crucial for preparing students to navigate the complexities of the modern world. Critical thinking, project-based learning, and volunteering enhance academic learning and empower students to address real-world challenges effectively. By focusing on these aspects, students can develop the skills they need to innovate, collaborate, and positively impact their communities.

Critical Thinking

Critical thinking is a fundamental skill for problem-solving as it involves analysing and evaluating information to make reasoned judgments and decisions. It enables students to approach problems systematically, consider multiple perspectives, and identify underlying issues.

Critical thinking allows students to:

• Analyse information : Students can assess the relevance and reliability of information to determine its impact on problem-solving. For example, in a science project, critical thinking helps students evaluate experimental results to draw valid conclusions.
• Develop solutions : Students can choose the most effective solution by critically evaluating different approaches. In a group project, critical thinking enables students to compare and refine ideas to solve a problem creatively.

Project-Based Learning

Project-based learning (PBL) is an instructional approach where students learn by actively engaging in real-world and personally meaningful projects. It allows students to explore complex problems and develop essential skills such as collaboration and communication.

Here is how project-based learning helps students develop problem-solving skills.

• Apply knowledge : Students apply academic concepts to real-world problems by working on projects. For instance, in designing a community garden, students use math to plan the layout and science to understand plant growth.
• Develop skills : PBL fosters problem-solving by challenging students to address authentic problems. For example, in a history project, students might analyse primary sources to understand the causes of historical events and propose solutions to prevent similar conflicts.

Volunteering

Volunteering allows students to contribute to their communities while developing empathy, leadership , and problem-solving skills. It provides practical experiences that enhance learning and help students understand and address community needs.

Volunteering is important because it allows students to:

• Identify needs : Students can identify community needs and consider solutions by working in diverse settings. For example, volunteering at a food bank can inspire students to address food insecurity by organising donation drives.
• Collaborate : Volunteering encourages teamwork and collaboration to solve problems. Students learn to coordinate tasks and resources to achieve common goals when organising a charity event.

The Problem-Solving Process

Problem-solving involves a systematic approach to understanding, analysing, and solving problems. Here are the critical steps in the problem-solving process:

• Identify the problem : The first step is clearly defining and understanding the problem. This involves identifying the specific issue or challenge that needs to be addressed.
• Define goals : Once the problem is identified, it's essential to establish clear and measurable goals. This helps focus efforts and guide the problem-solving process.
• Explore possible solutions : The next step is brainstorming and exploring various solutions. This involves generating ideas and considering different approaches to solving the problem.
• Evaluate options : After generating potential solutions, evaluate each option based on its feasibility, effectiveness, and possible outcomes.
• Choose the best solution : Select the most appropriate solution that best meets the defined goals and addresses the root cause of the problem.
• Implement the solution : Once a solution is chosen, it must be implemented. This step involves planning the implementation process and taking necessary actions to execute the solution.
• Monitor progress : After implementing the solution, monitor its progress and evaluate its effectiveness. This step helps ensure that the problem is being resolved as expected.
• Reflect and adjust : Reflect on the problem-solving process, identify any lessons learned, and make adjustments if necessary. This continuous improvement cycle helps refine solutions and develop better problem-solving skills.

How to Become a General Problem Solver

Parents play a crucial role in nurturing their children's problem-solving skills. Here are some ways parents can help their children become effective problem solvers.

• Encourage critical thinking : Encourage children to ask questions, analyse information, and consider different perspectives. Engage them in discussions that challenge their thinking and promote reasoning.
• Support independence : Allow children to tackle challenges on their own. Offer guidance and encouragement without immediately providing solutions. This helps build confidence and resilience.
• Provide opportunities for problem-solving : Create opportunities for children to solve real-life problems, such as planning a family event, organising their room, or resolving conflicts with siblings or friends.
• Foster creativity : Encourage creative thinking and brainstorming. Provide materials and activities that stimulate imagination and innovation.
• Model problem-solving behaviours : Demonstrate problem-solving skills in your own life and involve children in decision-making processes. Show them how to approach challenges calmly and methodically.

How Online Schooling Encourages Problem-Solving

Online schooling encourages problem-solving skills by requiring students to navigate digital platforms, manage their time effectively , and troubleshoot technical issues independently. 

Students often engage in interactive assignments and projects that promote critical thinking and creativity. They learn to adapt to different learning environments and collaborate virtually, fostering innovative solutions. 

Online schooling also encourages self-directed learning , where students must identify and address their own learning gaps. This enhances problem-solving abilities and prepares them for the complexities of the digital age.

To find out more about online learning, click here . 

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• Problem Solving in STEM

Solving problems is a key component of many science, math, and engineering classes.  If a goal of a class is for students to emerge with the ability to solve new kinds of problems or to use new problem-solving techniques, then students need numerous opportunities to develop the skills necessary to approach and answer different types of problems.  Problem solving during section or class allows students to develop their confidence in these skills under your guidance, better preparing them to succeed on their homework and exams. This page offers advice about strategies for facilitating problem solving during class.

How do I decide which problems to cover in section or class?

In-class problem solving should reinforce the major concepts from the class and provide the opportunity for theoretical concepts to become more concrete. If students have a problem set for homework, then in-class problem solving should prepare students for the types of problems that they will see on their homework. You may wish to include some simpler problems both in the interest of time and to help students gain confidence, but it is ideal if the complexity of at least some of the in-class problems mirrors the level of difficulty of the homework. You may also want to ask your students ahead of time which skills or concepts they find confusing, and include some problems that are directly targeted to their concerns.

You have given your students a problem to solve in class. What are some strategies to work through it?

• Try to give your students a chance to grapple with the problems as much as possible.  Offering them the chance to do the problem themselves allows them to learn from their mistakes in the presence of your expertise as their teacher. (If time is limited, they may not be able to get all the way through multi-step problems, in which case it can help to prioritize giving them a chance to tackle the most challenging steps.)
• When you do want to teach by solving the problem yourself at the board, talk through the logic of how you choose to apply certain approaches to solve certain problems.  This way you can externalize the type of thinking you hope your students internalize when they solve similar problems themselves.
• Start by setting up the problem on the board (e.g you might write down key variables and equations; draw a figure illustrating the question).  Ask students to start solving the problem, either independently or in small groups.  As they are working on the problem, walk around to hear what they are saying and see what they are writing down. If several students seem stuck, it might be a good to collect the whole class again to clarify any confusion.  After students have made progress, bring the everyone back together and have students guide you as to what to write on the board.
• It can help to first ask students to work on the problem by themselves for a minute, and then get into small groups to work on the problem collaboratively.
• If you have ample board space, have students work in small groups at the board while solving the problem.  That way you can monitor their progress by standing back and watching what they put up on the board.
• If you have several problems you would like to have the students practice, but not enough time for everyone to do all of them, you can assign different groups of students to work on different – but related - problems.

When do you want students to work in groups to solve problems?

• Don’t ask students to work in groups for straightforward problems that most students could solve independently in a short amount of time.
• Do have students work in groups for thought-provoking problems, where students will benefit from meaningful collaboration.
• Even in cases where you plan to have students work in groups, it can be useful to give students some time to work on their own before collaborating with others.  This ensures that every student engages with the problem and is ready to contribute to a discussion.

What are some benefits of having students work in groups?

• Students bring different strengths, different knowledge, and different ideas for how to solve a problem; collaboration can help students work through problems that are more challenging than they might be able to tackle on their own.
• In working in a group, students might consider multiple ways to approach a problem, thus enriching their repertoire of strategies.
• Students who think they understand the material will gain a deeper understanding by explaining concepts to their peers.

What are some strategies for helping students to form groups?  

• Instruct students to work with the person (or people) sitting next to them.
• Count off.  (e.g. 1, 2, 3, 4; all the 1’s find each other and form a group, etc)
• Hand out playing cards; students need to find the person with the same number card. (There are many variants to this.  For example, you can print pictures of images that go together [rain and umbrella]; each person gets a card and needs to find their partner[s].)
• Based on what you know about the students, assign groups in advance. List the groups on the board.
• Note: Always have students take the time to introduce themselves to each other in a new group.

What should you do while your students are working on problems?

• Walk around and talk to students. Observing their work gives you a sense of what people understand and what they are struggling with. Answer students’ questions, and ask them questions that lead in a productive direction if they are stuck.
• If you discover that many people have the same question—or that someone has a misunderstanding that others might have—you might stop everyone and discuss a key idea with the entire class.

After students work on a problem during class, what are strategies to have them share their answers and their thinking?

• Ask for volunteers to share answers. Depending on the nature of the problem, student might provide answers verbally or by writing on the board. As a variant, for questions where a variety of answers are relevant, ask for at least three volunteers before anyone shares their ideas.
• Use online polling software for students to respond to a multiple-choice question anonymously.
• If students are working in groups, assign reporters ahead of time. For example, the person with the next birthday could be responsible for sharing their group’s work with the class.
• Cold call. To reduce student anxiety about cold calling, it can help to identify students who seem to have the correct answer as you were walking around the class and checking in on their progress solving the assigned problem. You may even want to warn the student ahead of time: "This is a great answer! Do you mind if I call on you when we come back together as a class?"
• Have students write an answer on a notecard that they turn in to you.  If your goal is to understand whether students in general solved a problem correctly, the notecards could be submitted anonymously; if you wish to assess individual students’ work, you would want to ask students to put their names on their notecard.  
• Use a jigsaw strategy, where you rearrange groups such that each new group is comprised of people who came from different initial groups and had solved different problems.  Students now are responsible for teaching the other students in their new group how to solve their problem.
• Have a representative from each group explain their problem to the class.
• Have a representative from each group draw or write the answer on the board.

What happens if a student gives a wrong answer?

• Ask for their reasoning so that you can understand where they went wrong.
• Ask if anyone else has other ideas. You can also ask this sometimes when an answer is right.
• Cultivate an environment where it’s okay to be wrong. Emphasize that you are all learning together, and that you learn through making mistakes.
• Do make sure that you clarify what the correct answer is before moving on.
• Once the correct answer is given, go through some answer-checking techniques that can distinguish between correct and incorrect answers. This can help prepare students to verify their future work.

How can you make your classroom inclusive?

• The goal is that everyone is thinking, talking, and sharing their ideas, and that everyone feels valued and respected. Use a variety of teaching strategies (independent work and group work; allow students to talk to each other before they talk to the class). Create an environment where it is normal to struggle and make mistakes.
• See Kimberly Tanner’s article on strategies to promoste student engagement and cultivate classroom equity. 

A few final notes…

• Make sure that you have worked all of the problems and also thought about alternative approaches to solving them.
• Board work matters. You should have a plan beforehand of what you will write on the board, where, when, what needs to be added, and what can be erased when. If students are going to write their answers on the board, you need to also have a plan for making sure that everyone gets to the correct answer. Students will copy what is on the board and use it as their notes for later study, so correct and logical information must be written there.

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Tipsheet: Problem Solving in STEM Sections

Tanner, K. D. (2013). Structure matters: twenty-one teaching strategies to promote student engagement and cultivate classroom equity . CBE-Life Sciences Education, 12(3), 322-331.

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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. 

Problem-Solving Method in Teaching

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

Table of Contents

Definition of problem-solving method.

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

Meaning of Problem-Solving Method

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

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

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

Benefits of Problem-Solving Method

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

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

Steps in Problem-Solving Method

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

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

Verification of the concluded solution or Hypothesis

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

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

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

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

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

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

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

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

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

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

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

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

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

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

Key features of a good mathematics problem includes:

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

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

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

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

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

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

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

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

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

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

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

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

Number Talks

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

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

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

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

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

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

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

Instead, you and your students could say the following:

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

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

Using “Worksheets”

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

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

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

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

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

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

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

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

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

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

when we teach students how to problem solve

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

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

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

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

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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.

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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

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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.

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How to Choose: Let’s Draw a Comparison

The following table compares the different problem-solving methods:

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.

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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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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.

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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.

Benefits of Problem-Solving in the K-12 Classroom

Posted October 5, 2022 by Miranda Marshall

From solving complex algebra problems to investigating scientific theories, to making inferences about written texts, problem-solving is central to every subject explored in school. Even beyond the classroom, problem-solving is ranked among the most important skills for students to demonstrate on their resumes, with 82.9% of employers considering it a highly valued attribute. On an even broader scale, students who learn how to apply their problem-solving skills to the issues they notice in their communities – or even globally –  have the tools they need to change the future and leave a lasting impact on the world around them.

Problem-solving can be taught in any content area and can even combine cross-curricular concepts to connect learning from all subjects. On top of building transferrable skills for higher education and beyond, read on to learn more about five amazing benefits students will gain from the inclusion of problem-based learning in their education:

• Problem-solving is inherently student-centered.

Student-centered learning refers to methods of teaching that recognize and cater to students’ individual needs. Students learn at varying paces, have their own unique strengths, and even further, have their own interests and motivations – and a student-centered approach recognizes this diversity within classrooms by giving students some degree of control over their learning and making them active participants in the learning process.

Incorporating problem-solving into your curriculum is a great way to make learning more student-centered, as it requires students to engage with topics by asking questions and thinking critically about explanations and solutions, rather than expecting them to absorb information in a lecture format or through wrote memorization.

• Increases confidence and achievement across all school subjects.

As with any skill, the more students practice problem-solving, the more comfortable they become with the type of critical and analytical thinking that will carry over into other areas of their academic careers. By learning how to approach concepts they are unfamiliar with or questions they do not know the answers to, students develop a greater sense of self-confidence in their ability to apply problem-solving techniques to other subject areas, and even outside of school in their day-to-day lives.

The goal in teaching problem-solving is for it to become second nature, and for students to routinely express their curiosity, explore innovative solutions, and analyze the world around them to draw their own conclusions.

• Encourages collaboration and teamwork.

Since problem-solving often involves working cooperatively in teams, students build a number of important interpersonal skills alongside problem-solving skills. Effective teamwork requires clear communication, a sense of personal responsibility, empathy and understanding for teammates, and goal setting and organization – all of which are important throughout higher education and in the workplace as well.

• Increases metacognitive skills.

Metacognition is often described as “thinking about thinking” because it refers to a person’s ability to analyze and understand their own thought processes. When making decisions, metacognition allows problem-solvers to consider the outcomes of multiple plans of action and determine which one will yield the best results.

Higher metacognitive skills have also widely been linked to improved learning outcomes and improved studying strategies. Metacognitive students are able to reflect on their learning experiences to understand themselves and the world around them better.

• Helps with long-term knowledge retention.

Students who learn problem-solving skills may see an improved ability to retain and recall information. Specifically, being asked to explain how they reached their conclusions at the time of learning, by sharing their ideas and facts they have researched, helps reinforce their understanding of the subject matter.

Problem-solving scenarios in which students participate in small-group discussions can be especially beneficial, as this discussion gives students the opportunity to both ask and answer questions about the new concepts they’re exploring.

At all grade levels, students can see tremendous gains in their academic performance and emotional intelligence when problem-solving is thoughtfully planned into their learning.

Interested in helping your students build problem-solving skills, but aren’t sure where to start? Future Problem Solving Problem International (FPSPI) is an amazing academic competition for students of all ages, all around the world, that includes helpful resources for educators to implement in their own classrooms!

Learn more about this year’s competition season from this recorded webinar:    https://youtu.be/AbeKQ8_Sm8U and/or email [email protected] to get started!

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Elementary teachers’ experience of engaging with Teaching Through Problem Solving using Lesson Study

Mairéad hourigan.

Department of STEM Education, Mary Immaculate College, University of Limerick, Limerick, Ireland

Aisling M. Leavy

For many decades, problem solving has been a focus of elementary mathematics education reforms. Despite this, in many education systems, the prevalent approach to mathematics problem solving treats it as an isolated activity instead of an integral part of teaching and learning. In this study, two mathematics teacher educators introduced 19 Irish elementary teachers to an alternative problem solving approach, namely Teaching Through Problem Solving (TTP), using Lesson Study (LS) as the professional development model. The findings suggest that the opportunity to experience TTP first-hand within their schools supported teachers in appreciating the affordances of various TTP practices. In particular, teachers reported changes in their beliefs regarding problem solving practice alongside developing problem posing knowledge. Of particular note was teachers’ contention that engaging with TTP practices through LS facilitated them to appreciate their students’ problem solving potential to the fullest extent. However, the planning implications of the TTP approach presented as a persistent barrier.

Introduction

A fundamental goal of mathematics education is to develop students’ ability to engage in mathematical problem solving. Despite curricular emphasis internationally on problem solving, many teachers are uncertain how to harness students’ problem solving potential (Cheeseman, 2018 ). While many problem solving programmes focus on providing students with step-by-step supports through modelling, heuristics, and other structures (Polya, 1957 ), Goldenberg et al. ( 2001 ) suggest that the most effective approach to developing students’ problem solving ability is by providing them with frequent opportunities over a prolonged period to solve worthwhile open-ended problems that are challenging yet accessible to all. This viewpoint is in close alignment with reform mathematics perspectives that promote conceptual understanding, where students actively construct their knowledge and relate new ideas to prior knowledge, creating a web of connected knowledge (Hiebert, 2003 ; Lester, 2013 ; Takahashi, 2006 ; Watanabe, 2001 ).

There is consensus in the mathematics education community that problem solving should not be taught as an isolated topic focused solely on developing problem solving skills and strategies or presented as an end-of-chapter activity (Takahashi, 2006 , 2016 ; Takahashi et al., 2013 ). Instead, problem solving should be integrated across the curriculum as a fundamental part of mathematics teaching and learning (Cai & Lester, 2010 ; Takahashi, 2016 ).

A ‘Teaching Through Problem Solving’ (TTP) approach, a problem solving style of instruction that originated in elementary education in Japan, meets these criteria treating problem solving as a core practice rather than an ‘add-on’ to mathematics instruction.

Teaching Through Problem Solving (TTP)

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 understand what is required. Students then solve the problem either individually or in groups, inventing their approaches. At this stage, the teacher does not model or suggest a solution procedure. Instead, they take on the role of facilitator, providing support to students only at the right time (Hiebert, 2003 ; Lester, 2013 ; Takahashi, 2006 ). As students solve the problem, the teacher circulates, observes the range of student strategies, and identifies work that illustrates desired features. However, the problem solving lesson does not end when the students find a solution. The subsequent sharing phase, called Neriage (polishing ideas), is considered by Japanese teachers to be the heart of the lesson rather than its culmination. During Neriage, the teacher purposefully selects students to share their strategies, compares various approaches, and introduces increasingly sophisticated solution methods. Effective questioning is central to this process, alongside careful recording of the multiple solutions on the board. The teacher concludes the lesson by formalising and consolidating the lesson’s main points. This process promotes learning for all students (Hiebert, 2003 ; Stacey, 2018 ; Takahashi, 2016 ; Takahashi et al., 2013 ; Watanabe, 2001 ).

The TTP approach assumes that students develop, extend, and enrich their understandings as they confront problematic situations using existing knowledge. Therefore, TTP fosters the symbiotic relationship between conceptual understanding and problem solving, as conceptual understanding is required to solve challenging problems and make sense of new ideas by connecting them with existing knowledge. Equally, problem solving promotes conceptual understanding through the active construction of knowledge (Hiebert, 2003 ; Lambdin, 2003 ; Takahashi, 2006 ). Consequently, students simultaneously develop more profound understandings of the mathematics content while cultivating problem solving skills (Kapur, 2010 ; Stacey, 2018 ).

Relevant research affirms that teachers acknowledge the merits of this approach (Sullivan et al., 2014 ) and most students report positive experiences (Russo & Minas, 2020 ). The process is considered to make students’ thinking and learning visible (Ingram et al., 2020 ). Engagement in TTP has resulted in teachers becoming more aware of and confident in their students’ problem solving abilities and subsequently expecting more from them (Crespo & Featherstone, 2006 ; Sakshaug & Wohlhuter, 2010 ).

Demands of TTP

Adopting a TTP approach challenges pre-existing beliefs and poses additional knowledge demands for elementary teachers, both content and pedagogical (Takahashi, 2008 ).

Research has consistently reported a relationship between teacher beliefs and the instructional techniques used, with evidence of more rule-based, teacher-directed strategies used by teachers with traditional mathematics beliefs (Stipek et al., 2001 ; Swan, 2006 ; Thompson, 1985 ). These teachers tend to address problem solving separately from concept and skill development and possess a simplistic view of problem solving as translating a problem into abstract mathematical terms to solve it. Consequently, such teachers ‘are very concerned about developing skilfulness in translating (so-called) real-world problems into mathematical representations and vice versa’ (Lester, 2013 , p. 254). Early studies of problem solving practice reported direct instructional techniques where the teacher would model how to solve the problem followed by students practicing similar problems (Chapman, 2015 ; Hiebert, 2003 ; Lester, 2013 ). This naïve conception of problem solving is reflected in many textbook problems that simply require students to apply previously learned routine procedures to solve problems that are merely thinly disguised number operations (Lester, 2013 ; Singer & Voica, 2013 ). Hence, the TTP approach requires a significant shift for teachers who previously considered problem solving as an extra activity conducted after the new mathematics concepts are introduced (Lester, 2013 ; Takahashi et al., 2013 ) or whose personal experience of problem solving was confined to applying routine procedures to word problems (Sakshaug &Wohlhuter, 2010 ).

Alongside beliefs, teachers’ knowledge influences their problem solving practices. Teachers require a deep understanding of the nature of problem solving, in particular viewing problem solving as a process (Chapman, 2015 ). To be able to understand the stages problem solvers go through and appreciate what successful problem solving involves, teachers benefit from experiencing solving problems from the problem solver’s perspective (Chapman, 2015 ; Lester, 2013 ).

It is also essential that teachers understand what constitutes a worthwhile problem when selecting or posing problems (Cai, 2003 ; Chapman, 2015 ; Lester, 2013 ; O’Shea & Leavy, 2013 ). This requires an understanding that problems are ‘mathematical tasks for which the student does not have an obvious way to solve it’ (Chapman, 2015 , p. 22). Teachers need to appreciate the variety of problem characteristics that contribute to the richness of a problem, e.g. problem structures and cognitive demand (Klein & Leiken, 2020 ; O’Shea & Leavy, 2013 ). Such understandings are extensive, and rather than invest heavily in the time taken to construct their mathematics problems, teachers use pre-made textbook problems or make cosmetic changes to make cosmetic changes to these (Koichu et al., 2013 ). In TTP, due consideration must also be given to the problem characteristics that best support students in strengthening existing understandings and experiencing new learning of the target concept, process, or skill (Cai, 2003 ; Takahashi, 2008 ). Specialised content knowledge is also crucial for teachers to accurately predict and interpret various solution strategies and misconceptions/errors, to determine the validity of alternative approaches and the source of errors, to sequence student approaches, and to synthesise approaches and new learning during the TTP lesson (Ball et al., 2008 ; Cai, 2003 ; Leavy & Hourigan, 2018 ).

Teachers should also be knowledgeable regarding appropriate problem solving instruction. It is common for teachers to teach for problem solving (i.e., focusing on developing students’ problem solving skills and strategies). Teachers adopting a TTP approach engage in reform classroom practices that reflect a constructivist-oriented approach to problem solving instruction where the teacher guides students to work collaboratively to construct meaning, deciding when and how to support students without removing their autonomy (Chapman, 2015 ; Hiebert, 2003 ; Lester, 2013 ). Teachers ought to be aware of the various relevant models of problem solving, including Polya’s ( 1957 ) model that supports teaching for problem solving (Understand the problem-Devise a plan-Carry out the plan-Look back) alongside models that support TTP (e.g., Launch-Explore-Summarise) (Lester, 2013 ; Sullivan et al., 2021 ). While knowledge of heuristics and strategies may support teachers’ problem solving practices, there is consensus that teaching heuristics and strategies or teaching about problem solving does not significantly improve students’ problem solving ability. Teachers require a thorough knowledge of their students as problem solvers, for example, being aware of their abilities and factors that hinder their success, including language (Chapman, 2015 ). Knowledge of content and student, alongside content and teaching (Ball et al., 2008 ), is essential during TTP planning when predicting student approaches and errors. Such knowledge is also crucial during TTP implementation when determining the validity of alternative approaches, identifying the source of errors (Explore phase), sequencing student approaches, and synthesising the range of approaches and new learning effectively (Summarise phase) (Cai, 2003 ; Leavy & Hourigan, 2018 ).

Supports for teachers

Given the extensive demands of TTP, adopting this approach is arduous in terms of the planning time required to problem pose, predict approaches, and design questions and resources (Lester, 2013 ; Sullivan et al., 2010 ; Takahashi,  2008 ). Consequently, it is necessary to support teachers who adopt a TTP approach (Hiebert, 2003 ). Professional development must facilitate them to experience the approach themselves as learners and then provide classroom implementation opportunities that incorporate collaborative planning and reflection when trialling the approach (Watanabe, 2001 ). In Japan, a common form of professional development to promote, develop, and refine TTP implementation among teachers and test potential problems for TTP is Japanese Lesson Study (LS) (Stacey, 2018 ; Takahashi et al., 2013 ). Another valuable support is access to a repository of worthwhile problems. In Japan, government-authorised textbooks and teacher manuals provide a sequence of lessons with rich well-tested problems to introduce new concepts. They also detail alternative strategies used by students and highlight the key mathematical aspects of these strategies (Takahashi, 2016 ; Takahashi et al., 2013 ).

Teachers’ reservations about TTP

Despite the acknowledged benefits of TTP for students, some teachers report reluctance to employ TTP, identifying a range of obstacles. These include limited mathematics content knowledge or pedagogical content knowledge (Charalambous, 2008 ; Sakshaug & Wohlhuter, 2010 ) and a lack of access to resources or time to develop or modify appropriate resources (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ). Other barriers for teachers with limited experience of TTP include giving up control, struggling to support students without directing them, and a tendency to demonstrate how to solve the problem (Cheeseman, 2018 ; Crespo & Featherstone, 2006 ; Klein & Leiken, 2020 ; Takahashi et al., 2013 ). Resistance to TTP is also associated with some teachers’ perception that this approach would lead to student disengagement and hence be unsuitable for lower-performing students (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2010 ).

Problem solving practices in Irish elementary mathematics education

Within the Irish context, problem solving is a central tenet of elementary mathematics curriculum documents (Department of Education and Science (DES), 1999 ) with recommendations that problem solving should be integral to students’ mathematical learning. However, research reveals a mismatch between intended and implemented problem solving practices (Dooley et al., 2014 ; Dunphy et al., 2014 ), where classroom practices reflect a narrow approach limited to problem solving as an ‘add on’, only applied after mathematical procedures had been learned and where problems are predominantly sourced from dedicated sections of textbooks (Department of Education and Skills (DES), 2011 ; Dooley et al., 2014 ; National Council for Curriculum and Assessment (NCCA), 2016 ; O’Shea & Leavy, 2013 ). Regarding the attained curriculum, Irish students have underperformed in mathematical problem solving, relative to other skills, in national and international assessments (NCCA, 2016 ; Shiel et al., 2014 ). Consensus exists that there is scope for improvement of problem solving practices, with ongoing calls for Irish primary teachers to receive support through school-based professional development models alongside creating a repository of quality problems (DES, 2011 ; Dooley et al., 2014 ; NCCA, 2016 ).

Lesson Study (LS) as a professional development model

Reform mathematics practices, such as TTP, challenge many elementary teachers’ beliefs, knowledge, practices, and cultural norms, particularly if they have not experienced the approach themselves as learners. To support teachers in enacting reform approaches, they require opportunities to engage in extended and targeted professional development involving collaborative and practice-centred experiences (Dudley et al., 2019 ; Murata et al., 2012 ; Takahashi et al., 2013 ). Lesson Study (LS) possesses the characteristics of effective professional development as it embeds ‘…teachers’ learning in their everyday work…increasing the likelihood that their learning will be meaningful’ (Fernandez et al., 2003 , p. 171).

In Japan, LS was developed in the 1980s to support teachers to use more student-centred practices. LS is a school-based, collaborative, reflective, iterative, and research-based form of professional development (Dudley et al., 2019 ; Murata et al., 2012 ). In Japan, LS is an integral part of teaching and is typically conducted as part of a school-wide project focused on addressing an identified teaching–learning challenge (Takahashi & McDougal, 2016 ). It involves a group of qualified teachers, generally within a single school, working together as part of a LS group to examine and better understand effective teaching practices. Within the four phases of the LS cycle, the LS group works collaboratively to study and plan a research lesson that addresses a pre-established goal before implementing (teach) and reflecting (observe, analyse and revise) on the impact of the lesson activities on students’ learning.

LS has become an increasingly popular professional development model outside of Japan in the last two decades. In these educational contexts, it is necessary to find a balance between fidelity to LS as originally envisaged and developing a LS approach that fits the cultural context of a country’s education system (Takahashi & McDougal, 2016 ).

Relevant research examining the impact of LS on qualified primary mathematics teachers reports many benefits. Several studies reveal that teachers demonstrated transformed beliefs regarding effective pedagogy and increased self-efficacy in their use due to engaging in LS (Cajkler et al., 2015 ; Dudley et al., 2019 ; Fernandez, 2005 ; Gutierez, 2016 ). Enhancements in participating teachers’ knowledge have also been reported (Cajkler et al., 2015 ; Dudley et al., 2019 ; Fernandez, 2005 ; Gutierez, 2016 ; Murata et al., 2012 ). Other gains recounted include improvements in practice with a greater focus on students (Cajkler et al., 2015 ; Dudley et al., 2019 ; Flanagan, 2021 ).

Context of this study

A cluster of urban schools, coordinated by their local Education Centre, engaged in an initiative to enhance teachers’ mathematics problem solving practices. The co-ordinator of the initiative approached the researchers, both mathematics teacher educators (MTEs), seeking a relevant professional development opportunity. Aware of the challenges of problem solving practice within the Irish context, the MTEs proposed an alternative perspective on problem solving: the Teaching Through Problem Solving (TTP) approach. Given Cai’s ( 2003 ) recommendation that teachers can best learn to teach through problem solving by teaching and reflecting as opposed to taking more courses, the MTEs identified LS as the best fit in terms of a supportive professional development model, as it is collaborative, experiential, and school-based (Dudley et al., 2019 ; Murata et al., 2012 ; Takahashi et al., 2013 ). Consequently, LS would promote teachers to work collaboratively to understand the TTP approach, plan TTP practices for their educational context, observe what it looks like in practice, and assess the impact on their students’ thinking (Takahashi et al., 2013 ). In particular, the MTEs believed that the LS phases and practices would naturally support TTP structures, emphasizing task selection and anticipating students’ solutions. Given Lester’s ( 2013 ) assertion that each problem solving experience a teacher engages in can potentially alter their knowledge for teaching problem solving, the MTEs sought to explore teachers’ perceptions of the impact of engaging with TTP through LS on their beliefs regarding problem solving and their knowledge for teaching problem solving.

Research questions

This paper examines two research questions:

• Research question 1: What are elementary teachers’ reported problem solving practices prior to engaging in LS?
• Research question 2: What are elementary teachers’ perceptions of what they learned from engaging with TTP through LS?

Methodology

Participants.

The MTEs worked with 19 elementary teachers (16 female, three male) from eight urban schools. Schools were paired to create four LS groups on the basis of the grade taught by participating class teachers, e.g. Grade 3 teacher from school 1 paired with Grade 4 teacher from school 2. Each LS group generally consisted of 4–5 teachers, with a minimum of two teachers from each school, along with the two MTEs. For most teachers, LS and TTP were new practices being implemented concurrently. However, given the acknowledged overlap between the features of the TTP and LS approaches, for example, the focus on problem posing and predicting student strategies, the researchers were confident that the content and structure were compatible. Also, in Japan, LS is commonly used to promote TTP implementation among teachers (Stacey, 2018 ; Takahashi et al., 2013 ).

All ethical obligations were adhered to throughout the research process, and the study received ethical approval from the researchers’ institutional board. Of the 19 participating LS teachers invited to partake in the research study, 16 provided informed consent to use their data for research purposes.

Over eight weeks, the MTEs worked with teachers, guiding each LS group through the four LS phases involving study, design, implementation, and reflection of a research lesson that focused on TTP while assuming the role of ‘knowledgeable others’ (Dudley et al., 2019 ; Hourigan & Leavy, 2021 ; Takahashi & McDougal, 2016 ). An overview of the timeline and summary of each LS phase is presented in Table ​ Table1 1 .

Lesson Study phases and timeline

LS phase 1: Study

This initial study phase involved a one-day workshop. The process and benefits of LS as a school-based form of professional development were discussed in the morning session and the afternoon component was spent focusing on the characteristics of TTP. Teachers experienced the TPP approach first-hand by engaging in the various lesson stages. For example, they solved a problem (growing pattern problem) themselves in pairs and shared their strategies. They also predicted children’s approaches to the problem and possible misconceptions and watched the video cases of TTP classroom practice for this problem. Particular focus was placed on the importance of problem selection and prediction of student strategies before the lesson implementation and the Neriage stage of the lesson. Teachers also discussed readings related to LS practices (e.g. Lewis & Tsuchida, 1998 ) and TTP (e.g., Takahashi, 2008 ). At the end of the workshop, members of each LS group were asked to communicate among themselves and the MTEs, before the planning phase, to decide the specific mathematics focus of their LS group’s TTP lesson (Table ​ (Table1 1 ).

LS phase 2: Planning

The planning phase was four weeks in duration and included two 1½ hour face-to-face planning sessions (i.e. planning meetings 1 and 2) between the MTEs and each LS group (Table ​ (Table1). 1 ). Meetings took place in one of the LS group’s schools. At the start of the first planning meeting, time was dedicated to Takahashi’s ( 2008 ) work focusing on the importance of problem selection and prediction of student strategies to plan the Neriage stage of the TTP lesson. The research lesson plan structure was also introduced. Ertle et al.’s ( 2001 ) four column lesson plan template was used. It was considered particularly compatible with the TTP approach, given the explicit attention to expected student response and the teacher’s response to student activity/response.

The planning then moved onto the content focus of each LS group’s TTP research lesson. LS groups selected TTP research lessons focusing on number (group A), growing patterns (group B), money (group C), and 3D shapes (group D). Across the planning phase, teachers invested substantial time extensively discussing the TTP lesson goals in terms of target mathematics content, developing or modifying a problem to address these goals, and exploring considerations for the various lesson stages. Drawing on Takahashi’s ( 2008 ) article, it was re-emphasised that no strategies would be explicitly taught before students engaged with the problem. While one LS group modified an existing problem (group B) (Hourigan & Leavy, 2015 ), the other three LS groups posed an original problem. To promote optimum teacher readiness to lead the Neriage stage, each LS group was encouraged to solve the problem themselves in various ways considering possible student strategies and their level of mathematical complexity, thus identifying the most appropriate sequence of sharing solutions.

LS phase 3: Implementation

The implementation phase involved one teacher in each LS group teaching the research lesson (teach 1) in their school. The remaining group members and MTEs observed and recorded students’ responses. Each LS group and the MTEs met immediately for a post-lesson discussion to evaluate the research lesson. The MTEs presented teachers with a series of focus questions: What were your observations of student learning? Were the goals of the lesson achieved? Did the problem support students in developing the appropriate understandings? Were there any strategies/errors that we had not predicted? How did the Neriage stage work? What aspects of the lesson plan should be reconsidered based on this evidence? Where appropriate, the MTEs drew teachers’ attention to particular lesson aspects they had not noticed. Subsequently, each LS group revised their research lesson in response to the observations, reflections, and discussion. The revised lesson was retaught 7–10 days later by a second group member from the paired LS group school (teach 2) (Table ​ (Table1). 1 ). The post-lesson discussion for teach 2 focused mainly on the impact of changes made after the first implementation on student learning, differences between the two classes, and further changes to the lesson.

LS phase 4: Reflection

While reflection occurred after both lesson implementations, the final reflection involved all teachers from the eight schools coming together for a half-day meeting in the local Education Centre to share their research lessons, experiences, and learning (Table ​ (Table1). 1 ). Each LS group made a presentation, identifying their research lesson’s content focus and sequence of activity. Artefacts (research lesson plan, materials, student work samples, photos) were used to support observations, reflections, and lesson modifications. During this meeting, teachers also reflected privately and in groups on their initial thoughts and experience of both LS and TTP, the benefits of participation, the challenges they faced, and they provided suggestions for future practice.

Data collection

The study was a collective case study (Stake, 1995 ). Each LS group constituted a case; thus, the analysis was structured around four cases. Data collection was closely aligned with and ran concurrent to the LS process. Table ​ Table2 2 details the links between the LS phases and the data collection process.

Overview of data collection methods across the research cycles

* MTE  mathematics teacher educator

The principal data sources (Table ​ (Table2) 2 ) included both MTEs’ fieldnotes (phase (P) 1–4), and reflections (P1–4), alongside email correspondence (P1–4), individual teacher reflections (P1, 2, 4) (see reflection tasks in Table ​ Table3), 3 ), and LS documentation including various drafts of lesson plans (P2–4) and group presentations (P4). Fieldnotes refer to all notes taken by MTEs when working with the LS groups, for example, during the study session, planning meetings, lesson implementations, post-lesson discussions, and the final reflection session.

Teacher reflection focus questions

The researchers were aware of the limitations of self-report data and the potential mismatch between one’s perceptions and reality. Furthermore, data in the form of opinions, attitudes, and beliefs may contain a certain degree of bias. However, this paper intentionally focuses solely on the teachers’ perceived learning in order to represent their ‘lived experience’ of TTP. Despite this, measures were taken to assure the trustworthiness and rigour of this qualitative study. The researchers engaged with the study over a prolonged period and collected data for each case (LS group) at every LS phase (Table ​ (Table2). 2 ). All transcripts reflected verbatim accounts of participants’ opinions and reflections. At regular intervals during the study, research meetings interrogated the researchers’ understandings, comparing participating teachers’ observations and reflections to promote meaning-making (Creswell, 2009 ; Suter, 2012 ).

Data analysis

The MTEs’ role as participant researchers was considered a strength of the research given that they possessed unique insights into the research context. A grounded theory approach was adopted, where the theory emerges from the data analysis process rather than starting with a theory to be confirmed or refuted (Glaser, 1978 ; Strauss & Corbin, 1998 ). Data were examined focusing on evidence of participants’ problem solving practices prior to LS and their perceptions of their learning as a result of engaging with TTP through LS. A systematic process of data analysis was adopted. Initially, raw data were organised into natural units of related data under various codes, e.g. resistance, traditional approach, ignorance, language, planning, fear of student response, relevance, and underestimation. Through successive examinations of the relationship between existing units, codes were amalgamated (Creswell, 2009 ). Progressive drafts resulted in the firming up of several themes. Triangulation was used to establish consistency across multiple data sources. While the first theme, Vast divide between prevalent problem solving practices and TTP , addresses research question 1, it is considered an overarching theme, given the impact of teachers’ established problem solving understandings and practices on their receptiveness to and experience of TTP. The remaining five themes ( Seeing is believing : the value of practice centred experiences ; A gained appreciation of the relevance and value of TTP practices ; Enhanced problem posing understandings ; Awakening to students’ problem solving potential ; and Reservations regarding TTP) represent a generalised model of teachers’ perceived learning due to engaging with TTP through LS, thus addressing research question 2. Although one of the researchers was responsible for the initial coding, both researchers met regularly during the analysis to discuss and interrogate the established codes and to agree on themes. This process served to counteract personal bias (Suter, 2012 ).

As teacher reflections were anonymised, it was not possible to track teachers across LS phases. Consequently, teacher reflection data are labelled as phase and instrument only. For example, ‘P2, teacher reflection’ communicates that the data were collected during LS phase 2 through teacher reflection. However, the remaining data are labelled according to phase, instrument, and source, e.g. ‘P3, fieldnotes: group B’. While phase 4 data reflect teachers’ perceptions after engaging fully with the TTP approach, data from the earlier phases reflect teachers’ evolving perceptions at a particular point in their unfolding TTP experience.

Discussion of findings

The findings draw on the analysis of the data collected across the LS phases and address the research questions. Within the confines of this paper, illustrative quotes are presented to provide insights into each theme. An additional layer of analysis was completed to ensure a balanced representation of teachers’ views in reporting findings. This process confirmed that the findings represent the views of teachers across LS groups, for example, within the first theme presented ( Vast divide ), the eight quotes used came from eight different teacher reflections. Equally, the six fieldnote excerpts selected represent six different teachers’ views across the four LS groups. Furthermore, in the second theme ( Seeing is believing ), the five quotes presented were sourced from five different participating teachers’ reflections and the six fieldnote excerpts included are from six different teachers across the four LS groups. Subsequent examination of the perceptions of those teachers not included in the reporting of findings confirmed that their perspectives were represented within the quotes used. Hence, the researchers are confident that the findings represent the views of teachers across all LS groups. For each theme, sources of evidence that informed the presented conclusions will be outlined.

Vast divide between prevalent problem solving practices and TTP

This overarching theme addresses the research question ‘What were elementary teachers’ reported problem solving practices prior to engaging in LS?’.

At the start of the initiative, within the study session (fieldnotes), all teachers identified mathematics problem solving as a problem of practice. The desire to develop problem solving practices was also apparent in some teachers’ reflections (phase 1 (P1), N  = 8):

I am anxious about it. Problem solving is an area of great difficulty throughout our school (P1, teacher reflection).

During both study and planning phase discussions, across all LS groups, teachers’ reports suggested the almost exclusive use of a teaching for problem solving approach, with no awareness of the Teaching Through Problem Solving (TTP) approach; a finding also evidenced in both teacher reflections (P1, N  = 7) and email correspondence:

Unfamiliar, not what I am used to. I have no experience of this kind of problem solving. This new approach is the reverse way to what I have used for problem solving (P1, teacher reflection) Being introduced to new methods of teaching problem solving and trying different approaches is both exciting and challenging (P1: email correspondence)

Teachers’ descriptions of their problem solving classroom practices in both teacher reflections (P1, N  = 8) and study session discussions (fieldnotes) suggested a naïve conception of problem solving, using heuristics such as the ‘RUDE (read, underline, draw a picture, estimate) strategy’ (P1, fieldnotes) to support students in decoding and solving the problem:

In general, the problem solving approach described by teachers is textbook-led, where concepts are taught context free first and the problems at the end of the chapter are completed afterwards (P1, reflection: MTE2)

This approach was confirmed as widespread across all LS groups within the planning meetings (fieldnotes).

In terms of problem solving instruction, a teacher-directed approach was reported by some teachers within teacher reflections (P1, N  = 5), where the teacher focused on a particular strategy and modelled its use by solving the problem:

I tend to introduce the problem, ensure everyone understands the language and what is being asked. I discuss the various strategies that children could use to solve the problem. Sometimes I demonstrate the approach. Then children practice similar problems … (P1: Teacher reflection)

However, it was evident within the planning meetings, that this traditional approach to problem solving was prevalent among the teachers in all LS groups. During the study session (field notes and teacher reflections (P1, N  = 7)), there was a sense that problem solving was an add-on as opposed to an integral part of mathematics teaching and learning. Again, within the planning meetings, discussions across all four LS groups verified this:

Challenge: Time to focus on problems not just computation (P1: Teacher reflection). From our discussions with the various LS groups’ first planning meeting, text-based teaching seems to be resulting in many teachers teaching concepts context-free initially and then matching the concept with the relevant problems afterwards (P2, reflection: MTE1)

However, while phase 1 teacher reflections suggested that a small number of participating teachers ( N  = 4) possessed broader problem solving understandings, subsequently during the planning meetings, there was ample evidence (field notes) of problem-posing knowledge and the use of constructivist-oriented approaches that would support the TTP approach among some participating teachers in each of the LS groups:

Challenge: Spend more time on meaningful problems and give them opportunities and time to engage in activities, rather than go too soon into tricks, rhymes etc (P1, Teacher reflection). The class are already used to sharing strategies and explaining where they went wrong (P2, fieldnotes, Group B) Teacher: The problem needs to have multiple entry points (P2, fieldnotes: Group C)

While a few teachers reported problem posing practices, in most cases, this consisted of cosmetic adjustments to textbook problems. Overall, despite evidence of some promising practices, the data evidenced predominantly traditional problem solving views and practices among participating teachers, with potential for further broadening of various aspects of their knowledge for teaching problem solving including what constitutes a worthwhile problem, the role of problem posing within problem solving, and problem solving instruction. Within phase 1 teacher reflections, when reporting ‘challenges’ to problem solving practices (Table ​ (Table3), 3 ), a small number of responses ( N  = 3) supported these conclusions:

Differences in teachers’ knowledge (P1: Teacher reflection). Need to challenge current classroom practices (P1: Teacher reflection).

However, from the outset, all participating teachers consistently demonstrated robust knowledge of their students as problem solvers, evidenced in phase 1 teacher reflections ( N  = 10) and planning meeting discussions (P2, fieldnotes). However, in these early phases, teachers generally portrayed a deficit view, focusing almost exclusively on the various challenges impacting their students’ problem solving abilities. While all teachers agreed that the language of problems was inhibiting student engagement, other common barriers reported included student motivation and perseverance:

They often have difficulties accessing the problem – they don’t know what it is asking them (P2, fieldnotes: Group C) Sourcing problems that are relevant to their lives. I need to change every problem to reference soccer so the children are interested (P1: teacher reflection) Our children deal poorly with struggle and are slow to consider alternative strategies (P2, fieldnotes: Group D)

Despite showcasing a strong awareness of their students’ problem solving difficulties, teachers initially demonstrated a lack of appreciation of the benefits accrued from predicting students’ approaches and misconceptions relating to problem solving. While it came to the researchers’ attention during the study phase, its prevalence became apparent during the initial planning meeting, as its necessity and purpose was raised in three of the LS groups:

What are the benefits of predicting the children’s responses? (P1, fieldnotes). I don’t think we can predict- we will have to wait and see (P2, fieldnotes: Group A).

This finding evidences teachers’ relatively limited knowledge for teaching problem solving, given that this practice is fundamental to TTP and constructivist-oriented approaches to problem solving instruction.

Perceived impacts of engaging with TTP through LS

In response to the research question ‘What are elementary teachers’ perceptions of what they learned from engaging with TTP through LS?’, thematic data analysis identified 5 predominant themes, namely, Seeing is believing : the value of practice centred experiences ; A gained appreciation of the relevance and value of TTP practices ; Enhanced problem posing understandings ; Awakening to students’ problem solving potential ; and Reservations regarding TTP.

Seeing is believing: the value of practice centred experiences

Teachers engaged with TTP during the study phase as both learners and teachers when solving the problem. They were also involved in predicting and analysing student responses when viewing the video cases, and engaged in extensive reading, discussion, and planning for their selected TTP problem within the planning phase. Nevertheless, teachers reported reservations about the relevance of TTP for their context within both phase 2 teacher reflections ( N  = 5) as well as within the planning meeting discourse of all LS groups. Teachers’ keen awareness of their students’ problem solving challenges, coupled with the vast divide between the nature of their prior problem solving practices and the TTP approach, resulted in teachers communicating concern regarding students’ possible reaction during the planning phase:

I am worried about the problem. I am concerned that if the problem is too complex the children won’t respond to it (P2, fieldnotes: Group B) The fear that the children will not understand the lesson objective. Will they engage? (P2, Teacher reflection)

Acknowledging their apprehension regarding students’ reactions to TTP, from the outset, all participating teachers communicated a willingness to trial TTP practices:

Exciting to be part of. Eager to see how it will pan out and the learning that will be taken from it (P1, teacher reflection) They should be ‘let off’ (P2, fieldnotes: Group A).

It was only within the implementation phase, when teachers received the opportunity to meaningfully observe the TTP approach in their everyday work context, with their students, that they explicitly demonstrated an appreciation for the value of TTP practices. It was evident from teacher commentary across all LS groups’ post-lesson discussions (fieldnotes) as well as in teacher reflections (P4, N  = 10) that observing first-hand the high levels of student engagement alongside students’ capacity to engage in desirable problem solving strategies and demonstrate sought-after dispositions had affected this change:

Class teacher: They engaged the whole time because it was interesting to them. The problem is core in terms of motivation. It determines their willingness to persevere. Otherwise, it won’t work whether they have the skills or not (P3, fieldnotes: Group C) LS group member: The problem context worked really well. The children were all eager and persevered. It facilitated all to enter at their own level, coming up with ideas and using their prior knowledge to solve the problem. Working in pairs and the concrete materials were very supportive. It’s something I’d never have done before (P3, fieldnotes: Group A)

Although all teachers showcased robust knowledge of their students’ problem solving abilities prior to engaging in TTP, albeit with a tendency to focus on their difficulties and factors that inhibited them, teachers’ contributions during post-lesson discussions (fieldnotes) alongside teacher reflections (P4, N  = 9) indicate that observing TTP in action supported them in developing an appreciation of value of the respective TTP practices, particularly the role of prediction and observation of students’ strategies/misconceptions in making the students’ thinking more visible:

You see the students through the process (P3, fieldnotes: Group C) It’s rare we have time to think, to break the problem down, to watch and understand children’s ways of thinking/solving. It’s really beneficial to get a chance to re-evaluate the teaching methods, to edit the lesson, to re-teach (P4, teacher reflection)

Analysis of the range of data sources across the phases suggests that it was the opportunity to experience TTP in practice in their classrooms that provided the ‘proof of concept’:

I thought it wasn’t realistic but bringing it down to your own classroom it is relevant (P4, teacher reflection).

Hence from the teachers’ perspective, they witnessed the affordances of TTP practices in the implementation phase of the LS process.

A gained appreciation of the relevance and value of TTP practices

While during the early LS phases, teachers’ reporting suggested a view of problem solving as teaching to problem solve, data from both fieldnotes (phases 3 and 4) and teacher reflections (phase 4) demonstrate that all teachers broadened their understanding of problem solving as a result of engaging with TTP:

Interesting to turn lessons on their head and give students the chance to think, plan and come up with possible strategies and solutions (P4, Teacher reflection)

On witnessing the affordances of TTP first-hand in their own classrooms, within both teacher reflections (P4, N  = 12) and LS group presentations, the teachers consistently reported valuing these new practices:

I just thought the whole way of teaching was a good way, an effective way of teaching. Sharing and exploring more than one way of solving is vital (P4, teacher reflection) There is a place for it in the classroom. I will use aspects of it going forward (P4, fieldnotes: Group C)

In fact, teachers’ support for this problem solving approach was apparent in phase 3 during the initial post-lesson discussions. It was particularly notable when a visitor outside of the LS group who observed teach 1 challenged the approach, recommending the explicit teaching of strategies prior to engagement. A LS group member’s reply evidenced the group’s belief that TTP naturally exposes students to the relevant learning: ‘Sharing and questioning will allow students to learn more efficient strategies [other LS group members nodding in agreement]’ (P3, fieldnotes; Group A).

In turn, within phase 4 teacher reflections, teachers consistently acknowledged that engaging with TTP through LS had challenged their understandings about what constitutes effective problem solving instruction ( N  = 12). In both teacher reflections (P4, N  = 14) and all LS group presentations, teachers reported an increased appreciation of the benefits of adopting a constructivist-oriented approach to problem solving instruction. Equally for some, this was accompanied by an acknowledgement of a heightened awareness of the limitations of their previous practice :

Really made me re-think problem solving lesson structures. I tend to spoon-feed them …over-scaffold, a lot of teacher talk. … I need to find a balance… (P4, teacher reflection) Less is more, one problem can be the basis for an entire lesson (P4, teacher reflection)

What was unexpected, was that some teachers (P4, N  = 8) reported that engaging with TTP through LS resulted in them developing an increased appreciation of the value of problem solving and the need for more regular opportunities for students to engage in problem solving:

I’ve come to realise that problem solving is critical and it should be focused on more often. I feel that with regular exposure to problems they’ll come to love being problem solvers (P4, teacher reflection)

Enhanced problem posing understandings

In the early phases of LS, few teachers demonstrated familiarity with problem characteristics (P2 teacher reflection, N  = 5). However, there was growth in teachers’ understandings of what constitutes a worthwhile problem and its role within TTP within all LS groups’ post-lesson discussions and presentations (fieldnotes) and teacher reflections (P4, N  = 10):

I have a deepened understanding of how to evaluate a problem (P2, teacher reflection) It’s essential to find or create a good problem with multiple strategies and/or solutions as a springboard for a topic. It has to be relevant and interesting for the kids (P4, teacher reflection)

As early as the planning phase, a small group of teachers’ reflections ( N  = 2) suggested an understanding that problem posing is an important aspect of problem solving that merits significant attention:

It was extremely helpful to problem solve the problem (P2, teacher reflection)

However, during subsequent phases, this realisation became more mainstream, evident within all LS groups’ post-lesson discussions and presentations (fieldnotes) and teacher reflections (P4, N  = 12):

During the first planning meeting, I was surprised and a bit anxious that we would never get to having created a problem. In hindsight, this was time well spent as the problem was crucial (P4, teacher reflection) I learned the problem is key. We don’t spend enough time picking the problem (P4, fieldnotes: Group C).

Alongside this, in all LS groups’ dialogues during the post-lesson discussion and presentations (fieldnotes) and teacher reflections (P4, N  = 15), teachers consistently demonstrated an enhanced awareness of the interdependence between the quality of the problem and students’ problem solving behaviours:

Better perseverance if the problem is of interest to them (P4, teacher reflection) It was an eye-opener to me, relevance is crucial, when the problem context is relevant to them, they are motivated to engage and can solve problems at an appropriate level…They all wanted to present (P3, fieldnotes: Group C)

The findings suggest that engaging with TTP through LS facilitated participating teachers to develop an enhanced understanding of the importance of problem posing and in identifying the features of a good mathematics problem, thus developing their future problem posing capacity. In essence, the opportunity to observe the TTP practices in their classrooms stimulated an enhanced appreciation for the value of meticulous attention to detail in TTP planning.

Awakening to students’ problem solving potential

In the final LS phases, teachers consistently reported that engaging with TTP through LS provided the opportunity to see the students through the process , thus supporting them in examining their students’ capabilities more closely. Across post-lesson discussions and presentations (fieldnotes) and teacher reflections (P4, N  = 14), teachers acknowledged that engagement in core TTP practices, including problem posing, prediction of students’ strategies during planning, and careful observation of approaches during the implementation phase, facilitated them to uncover the true extent of their students’ problem solving abilities, heightening their awareness of students’ proficiency in using a range of approaches:

Class teacher: While they took a while to warm up, I am most happy that they failed, tried again and succeeded. They all participated. Some found a pattern, others used trial and error. Others worked backward- opening the cube in different ways. They said afterward ‘That was the best maths class ever’ (P3, fieldnotes: Group D) I was surprised with what they could do. I have learned the importance of not teaching strategies first. I need to pull back and let the children solve the problems their own way and leave discussing strategies to the end (P4, teacher reflection)

In three LS groups, class teachers acknowledged in the post-lesson discussion (fieldnotes) that engaging with TTP had resulted in them realising their previous underestimation of [some or all] of their students’ problem solving abilities . Teacher reflections (P4, N  = 8) and LS group presentations (fieldnotes) also acknowledged this reality:

I underestimated my kids, which is awful. The children surprised me with the way they approached the problem. In the future I need to focus on what they can do as much as what might hinder them…they are more able than we may think (P4, reflection)

In all LS groups, teachers reported that their heightened appreciation of students’ problem solving capacities promoted them to use a more constructivist-orientated approach in the future:

I learned to trust the students to problem solve, less scaffolding. Children can be let off to explore without so much teacher intervention (P3, fieldnotes: Group D)

Some teachers ( N  = 3) also acknowledged the affective benefits of TTP on students:

I know the students enjoyed sharing their different strategies…it was great for their confidence (P4, teacher reflection)

Interestingly, in contrast with teachers’ initial reservations, their experiential and school-based participation in TTP through LS resulted in a lessening of concern regarding the suitability of TTP practices for their students. Hence, this practice-based model supported teachers in appreciating the full extent of their students’ capacities as problem solvers.

Reservations regarding TTP

When introduced to the concept of TTP in the study session, one teacher quickly addressed the time implications:

It is unrealistic in the everyday classroom environment. Time is the issue. We don’t have 2 hours to prep a problem geared at the various needs (P1, fieldnotes)

Subsequently, across the initiative, during both planning meetings, the reflection session and individual reflections (P4, N  = 14), acknowledgements of the affordances of TTP practices were accompanied by questioning of its sustainability due to the excessive planning commitment involved:

It would be hard to maintain this level of planning in advance of the lesson required to ensure a successful outcome (P4, teacher reflection)

Given the extensive time dedicated to problem posing, solving, prediction, and design of questions as well as selection or creation of materials both during and between planning meetings, there was agreement in the reflection session (fieldnotes) and in teacher reflections (P4, N  = 10) that while TTP practices were valuable, in the absence of suitable support materials for teachers, adjustments were essential to promote implementation:

There is definitely a role for TTP in the classroom, however the level of planning involved would have to be reduced to make it feasible (P4, teacher reflection) The TTP approach is very effective but the level of planning involved is unrealistic with an already overcrowded curriculum. However, elements of it can be used within the classroom (P4, teacher reflection)

A few teachers ( N  = 3) had hesitations beyond the time demands, believing the success of TTP is contingent on ‘a number of criteria…’ (P4, teacher reflection):

A whole-school approach is needed, it should be taught from junior infants (P4, teacher reflection) I still have worries about TTP. We found it difficult to decide a topic initially. It lends itself to certain areas. It worked well for shape and space (P4, teacher reflection)

Conclusions

The reported problem solving practice reflects those portrayed in the literature (NCCA, 2016 ; O’Shea & Leavy, 2013 ) and could be aptly described as ‘pendulum swings between emphases on basic skills and problem solving’ (Lesh & Zawojewski, 2007 in Takahashi et al., 2013 , p. 239). Teachers’ accounts depicted problem solving as an ‘add on’ occurring on an ad hoc basis after concepts were taught (Dooley et al., 2014 ; Takahashi et al., 2013 ), suggesting a simplistic view of problem solving (Singer & Voica, 2013 ; Swan, 2006 ). Hence, in reality there was a vast divide between teachers’ problem solving practices and TTP. Alongside traditional beliefs and problem solving practices (Stipek et al., 2001 ; Swan, 2006 ; Thompson, 1985 ), many teachers demonstrated limited insight regarding what constitutes a worthwhile problem (Klein & Lieken, 2020 ) or the critical role of problem posing in problem solving (Cai, 2003 ; Takahashi, 2008 ; Watson & Ohtani, 2015 ). Teachers’ reports suggested most were not actively problem posing, with reported practices limited to cosmetic changes to the problem context (Koichu et al., 2013 ). Equally, teachers demonstrated a lack of awareness of alternative approaches to teaching for problem solving (Chapman, 2015 ) alongside limited appreciation among most of the affordances of a more child-centred approach to problem solving instruction (Hiebert, 2003 ; Lester, 2013 ; Swan, 2006 ). Conversely, there was evidence that some teachers held relevant problem posing knowledge and utilised practices compatible with the TTP approach.

All teachers displayed relatively strong understandings of their students as problem solvers from the outset; however, they initially focused almost exclusively on factors impacting students’ limited problem solving capacity (Chapman, 2015 ). Teachers’ perceptions of their students’ problem solving abilities alongside the vast divide between teachers’ problem solving practice and the proposed TTP approach resulted in teachers being initially concerned regarding students’ response to TTP. This finding supports studies that reported resistance by teachers to the use of challenging tasks due to fears that students would not be able to manage (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2010 ). Equally, teachers communicated disquiet from the study phase regarding the time investment required to adopt the TTP approach, a finding common in similar studies (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ). Hence, the transition to TTP was uneasy for most teachers, given the significant shift it represented in terms of moving beyond a teaching to problem solve approach alongside the range of teacher demands (Takahashi et al., 2013 ).

Nevertheless, despite initial reservations, all teachers reported that engagement with TTP through LS affected their problem solving beliefs and understandings. What was particularly notable was that they reported an awakening to students’ problem solving potential . During LS’s implementation and reflection stages, all teachers acknowledged that seeing was believing concerning the benefits of TTP for their students (Kapur, 2010 ; Stacey, 2018 ). In particular, they recognised students’ positive response (Russo & Minas, 2020 ) enacted in high levels of engagement, perseverance in finding a solution, and the utilisation of a range of different strategies. These behaviours were in stark contrast to teachers’ reports in the study phase. Teachers acknowledged that students had more potential to solve problems autonomously than they initially envisaged. This finding supports previous studies where teachers reported that allowing students to engage with challenging tasks independently made students’ thinking more visible (Crespo & Featherstone, 2006 ; Ingram et al., 2020 ; Sakshand & Wohluter, 2010 ). It also reflects Sakshaug and Wohlhuter’s ( 2010 ) findings of teachers’ tendency to underestimate students’ potential to solve problems. Interestingly, at the end of LS, concern regarding the appropriateness of the TTP approach for students was no longer cited by teachers. This finding contrasts with previous studies that report teacher resistance due to fears that students will become disengaged due to the unsuitability of the approach (challenging tasks) for lower-performing students (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ). Hence, engaging with TTP through LS supported teachers in developing an appreciation of their students’ potential as problem solvers.

Teachers reported enhanced problem posing understandings, consisting of newfound awareness of the connections between the quality of the problem, the approach to problem solving instruction, and student response (Chapman, 2015 ; Cai, 2003 ; Sullivan et al., 2015 ; Takahashi, 2008 ). They acknowledged that they had learned the importance of the problem in determining the quality of learning and affecting student engagement, motivation and perseverance, and willingness to share strategies (Cai, 2003 ; Watson & Oktani, 2015 ). These findings reflect previous research reporting that engagement in LS facilitated teachers to enhance their teacher knowledge (Cajkler et al., 2015 ; Dudley et al., 2019 ; Gutierez, 2016 ).

While all teachers acknowledged the benefits of the TTP approach for students (Cai & Lester, 2010 ; Sullivan et al., 2014 ; Takahashi, 2016 ), the majority confirmed their perception of the relevance and value of various TTP practices (Hiebert, 2003 ; Lambdin, 2003 ; Takahashi, 2006 ). They referenced the benefits of giving more attention to the problem, allowing students the opportunity to independently solve, and promoting the sharing of strategies and pledged to incorporate these in their problem solving practices going forward. Many verified that the experience had triggered them to question their previous problem solving beliefs and practices (Chapman, 2015 ; Lester, 2013 ; Takahashi et al., 2013 ). This study supports previous research reporting that LS challenged teachers’ beliefs regarding the characteristics of effective pedagogy (Cajkler et al., 2015 ; Dudley et al., 2019 ; Fernandez, 2005 ; Gutierez, 2016 ). However, teachers communicated reservations regarding TTP , refraining from committing to TTP in its entirety, highlighting that the time commitment required for successful implementation on an ongoing basis was unrealistic. Therefore, teachers’ issues with what they perceived to be the excessive resource implications of TTP practices remained constant across the initiative. This finding supports previous studies that report teachers were resistant to engaging their students with ‘challenging tasks’ provided by researchers due to the time commitment required to plan adequately (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ).

Unlike previous studies, teachers in this study did not perceive weak mathematics content or pedagogical content knowledge as a barrier to implementing TTP (Charalambous, 2008 ; Sakshaug & Wohlhuter, 2010 ). However, it should be noted that the collaborative nature of LS may have hidden the knowledge demands for an individual teacher working alone when engaging in the ‘Anticipate’ element of TTP particularly in the absence of appropriate supports such as a bank of suitable problems.

The findings suggest that LS played a crucial role in promoting reported changes, serving both as a supportive professional development model (Stacey, 2018 ; Takahashi et al., 2013 ) and as a catalyst, providing teachers with the opportunity to engage in a collaborative, practice-centred experience over an extended period (Dudley et al., 2019 ; Watanabe, 2001 ). The various features of the LS process provided teachers with opportunities to engage with, interrogate, and reflect upon key TTP practices. Reported developments in understandings and beliefs were closely tied to meaningful opportunities to witness first-hand the affordances of the TTP approach in their classrooms with their students (Dudley et al., 2019 ; Fernandez et al., 2003 ; Takahashi et al., 2013 ). We suggest that the use of traditional ‘one-off’ professional development models to introduce TTP, combined with the lack of support during the implementation phase, would most likely result in teachers maintaining their initial views about the unsuitability of TTP practices for their students.

In terms of study limitations, given that all data were collected during the LS phases, the findings do not reflect the impact on teachers’ problem solving classroom practice in the medium to long term. Equally, while acknowledging the limitations of self-report data, there was no sense that the teachers were trying to please the MTEs, as they were forthright when invited to identify issues. Also, all data collected through teacher reflection was anonymous. The relatively small number of participating teachers means that the findings are not generalisable. However, they do add weight to the body of relevant research. This study also contributes to the field as it documents potential challenges associated with implementing TTP for the first time. It also suggests that despite TTP being at odds with their problem solving practice and arduous, the opportunity to experience the impact of the TTP approach with students through LS positively affected teachers’ problem solving understandings and beliefs and their commitment to incorporating TTP practices in their future practice. Hence, this study showcases the potential role of collaborative, school-based professional development in supporting the implementation of upcoming reform proposals (Dooley et al., 2014 ; NCCA, 2016 , 2017 , 2020 ), in challenging existing beliefs and practices and fostering opportunities for teachers to work collaboratively to trial reform teaching practices over an extended period (Cajkler et al., 2015 ; Dudley et al., 2019 ). Equally, this study confirms and extends previous studies that identify time as an immense barrier to TTP. Given teachers’ positivity regarding the impact of the TTP approach, their consistent acknowledgement of the unsustainability of the unreasonable planning demands associated with TTP strengthens previous calls for the development of quality support materials in order to avoid resistance to TTP (Clarke et al., 2014 ; Takahashi, 2016 ).

The researchers are aware that while the reported changes in teachers’ problem solving beliefs and understandings are a necessary first step, for significant and lasting change to occur, classroom practice must change (Sakshaug & Wohlhuter, 2010 ). While it was intended that the MTEs would work alongside interested teachers and schools to engage further in TTP in the school term immediately following this research and initial contact had been made, plans had to be postponed due to the commencement of the COVID 19 pandemic. The MTEs are hopeful that it will be possible to pick up momentum again and move this initiative to its natural next stage. Future research will examine these teachers’ perceptions of TTP after further engagement and evaluate the effects of more regular opportunities to engage in TTP on teachers’ problem solving practices. Another possible focus is teachers’ receptiveness to TTP when quality support materials are available.

In practical terms, in order for teachers to fully embrace TTP practices, thus facilitating their students to avail of the many benefits accrued from engagement, teachers require access to professional development (such as LS) that incorporates collaboration and classroom implementation at a local level. However, quality school-based professional development alone is not enough. In reality, a TTP approach cannot be sustained unless teachers receive access to quality TTP resources alongside formal collaboration time.

Acknowledgements

The authors acknowledge the participating teachers’ time and contribution to this research study.

This work was supported by the Supporting Social Inclusion and Regeneration in Limerick’s Programme Innovation and Development Fund.

Declarations

We have received ethical approval for the research presented in this manuscript from Mary Immaculate College Research Ethical Committee (MIREC).

The manuscript has only been submitted to Mathematics Education Research Journal. All authors have approved the manuscript submission. We also acknowledge that the submitted work is original and the content of the manuscript has not been published or submitted for publication elsewhere.

Informed consent has been received for all data included in this study. Of the 19 participating teachers, 16 provided informed consent.

The authors declare no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

• Ball DL, Thames MH, Phelps G. Content knowledge for teaching: What makes it special? Journal of Teacher Education. 2008; 59 (5):389–408. doi: 10.1177/0022487108324554. [ CrossRef ] [ Google Scholar ]
• Cai J. What research tells us about teaching mathematics through problem solving. In: Lester F, editor. Research and issues in teaching mathematics through problem solving. National Council for Curriculum and Assessment; 2003. pp. 241–255. [ Google Scholar ]
• Cai, J., & Lester, F. (2010). Why is teaching through problem solving important to children learning? National Council of Teachers of Mathematics.
• Cajkler W, Wood P, Norton J, Pedder D, Xu H. Teacher perspectives about lesson study in secondary school departments: A collaborative vehicle for professional learning and practice development. Research Papers in Education. 2015; 30 (2):192–213. doi: 10.1080/02671522.2014.887139. [ CrossRef ] [ Google Scholar ]
• Chapman O. Mathematics teachers’ knowledge for teaching problem solving. LUMAT International Journal on Math Science and Technology Education. 2015; 3 (1):19–36. doi: 10.31129/lumat.v3i1.1049. [ CrossRef ] [ Google Scholar ]
• Charalambous, C. Y. (2008). Mathematical knowledge for teaching and the unfolding of tasks in mathematics lessons: Integrating two lines of research. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Proceedings of the 32nd conference of the International Group for the Psychology of Mathematics Education (pp. 281–288). PME.
• Cheeseman, J. (2018). Teachers’ perceptions of obstacles to incorporating a problem solving style of mathematics into their teaching, In J. Hunter, P. Perger, & L. Darragh (Eds.), Making waves, opening spaces (Proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia) (pp. 210–217). MERGA.
• Clarke D, Cheeseman J, Roche A, Van Der Schans S. Teaching strategies for building student persistence on challenging tasks: Insights emerging from two approaches to teacher professional learning. Mathematics Teacher Education and Development. 2014; 16 (2):46–70. [ Google Scholar ]
• Creswell, J. W. (2009). Research design- Qualitative, quantitative and mixed methods approaches (3rd ed). Sage. [ PubMed ]
• Crespo S, Featherstone H. Teacher learning in mathematics teacher groups: One math problem at a time. In: Lynch-Davis K, Rider RL, editors. The work of mathematics teacher educators: Continuing the conversation. Association of Mathematics Teacher Educators; 2006. pp. 97–115. [ Google Scholar ]
• Department of Education and Science (DES). (1999). Primary school mathematics curriculum . The Stationery Office.
• Department of Education and Skills (DES). (2011). Literacy and numeracy for learning and life: The national strategy to improve literacy and numeracy among children and young people 2011–2020 . The Stationary Office.
• Dooley, T., Dunphy, E., & Shiel, G. (2014). Mathematics in early childhood and primary education. Research report 18. National Council for Curriculum and Assessment.
• Dudley P, Xu H, Vermunt JD, Lang J. Empirical evidence of the impact of lesson study on students’ achievement, teachers’ professional learning and on institutional and system evolution. European Journal of Education. 2019; 54 :202–217. doi: 10.1111/ejed.12337. [ CrossRef ] [ Google Scholar ]
• Dunphy, E., Dooley, T., Shiel, G. (2014). Mathematics in early childhood and primary education. Research report 17 . National Council for Curriculum and Assessment.
• Ertle, B., Chokshi, S., & Fernandez, C. (2001). Lesson planning tool. Available online: https://sarahbsd.files.wordpress.com/2014/09/lesson_planning_tool.pdf
• Fernandez C. Lesson Study: A means for elementary teachers to develop the knowledge of mathematics needed for reform-minded teaching? Mathematical Thinking and Learning. 2005; 7 (4):265–289. doi: 10.1207/s15327833mtl0704_1. [ CrossRef ] [ Google Scholar ]
• Fernandez C, Cannon J, Chokshi S. A US–Japan lesson study collaboration reveals critical lenses for examining practice. Teaching and Teacher Education. 2003; 19 :171–185. doi: 10.1016/S0742-051X(02)00102-6. [ CrossRef ] [ Google Scholar ]
• Flanagan, B. (2021). Teachers’ understandings of lesson study as a professional development tool, Unpublished thesis, University of Limerick.
• Glaser, B. G. (1978). Theoretical sensitivity: Advances in the methodology of grounded theory . Sociology Press.
• Goldenberg EP, Shteingold N, Feurzig N. Mathematical habits of mind for young children. In: Lester F, editor. Research and issues in teaching mathematics through problem solving. National Council for Curriculum and Assessment; 2001. pp. 15–30. [ Google Scholar ]
• Gutierez SB. Building a classroom-based professional learning community through lesson study: Insights from elementary school science teachers. Professional Development in Education. 2016; 42 (5):801–817. doi: 10.1080/19415257.2015.1119709. [ CrossRef ] [ Google Scholar ]
• Hiebert J. Signposts for teaching mathematics through problem solving. In: Lester F, editor. Research and issues in teaching mathematics through problem solving. National Council for Curriculum and Assessment; 2003. pp. 53–62. [ Google Scholar ]
• Hourigan, M., & Leavy, A. (2015). Geometric growing patterns: What’s the rule? Australian Primary Mathematics Classroom, 20 (4), 31–40.
• Hourigan, M., & Leavy, A. M. (2021). The complexities of assuming the ‘teacher of teachers’ role during Lesson Study.  Professional Development in Education. Online first. 10.1080/19415257.2021.1895287
• Ingram N, Holmes M, Linsell C, Livy S, McCormick M, Sullivan P. Exploring an innovative approach to teaching mathematics through the use of challenging tasks: A New Zealand perspective. Mathematics Education Research Journal. 2020; 32 (3):497–522. doi: 10.1007/s13394-019-00266-1. [ CrossRef ] [ Google Scholar ]
• Kapur M. Productive failure in mathematical problem solving. Instructional Science. 2010; 38 :523–550. doi: 10.1007/s11251-009-9093-x. [ CrossRef ] [ Google Scholar ]
• Klein S, Leikin R. Opening mathematical problems for posing open mathematical tasks: What do teachers do and feel? Educational Studies in Mathematics. 2020; 105 :349–365. doi: 10.1007/s10649-020-09983-y. [ CrossRef ] [ Google Scholar ]
• Koichu B, Harel G, Manaster A. Ways of thinking associated with mathematics teachers’ problem posing in the context of division of fractions. Instructional Science. 2013; 41 (4):681–698. doi: 10.1007/s11251-012-9254-1. [ CrossRef ] [ Google Scholar ]
• Lambdin DV. Benefits of teaching through problem solving. In: Lester F, editor. Research and issues in teaching mathematics through problem solving. National Council for Curriculum and Assessment; 2003. pp. 2–15. [ Google Scholar ]
• Leavy, A., & Hourigan, M. (2018). The role of perceptual similarity, data context and task context when selecting attributes: Examination of considerations made by 5-6 year olds in data modelling environments. Educational Studies in Mathematics. 97 (2), 163–183. 10.1007/s10649-017-9791-2
• Lesh R, Zawojewski J. Problem solving and modeling. In: Lester FK, editor. Second handbook of research on mathematics teaching and learning. Charlotte, NC: Information Age; 2007. pp. 763–804. [ Google Scholar ]
• Lester FK., Jr Thoughts about research on mathematical problem-solving instruction. The Mathematics Enthusiast. 2013; 10 (1–2):245–278. doi: 10.54870/1551-3440.1267. [ CrossRef ] [ Google Scholar ]
• Lewis C, Tsuchida I. A lesson is like a swiftly flowing river: How researc lessons improve Japanese ducation. American Educator. 1998; 22 (4):12–17. [ Google Scholar ]
• Murata A, Bofferding L, Pothen BE, Taylor MW, Wischnia S. Making connections among student learning, content, and teaching: Teacher talk paths in elementary mathematics lesson study. Journal for Research in Mathematics Education. 2012; 43 (5):616–650. doi: 10.5951/jresematheduc.43.5.0616. [ CrossRef ] [ Google Scholar ]
• National Council for Curriculum and Assessment (NCCA). (2016). Background paper and brief for the development of a new primary mathematics curriculum. NCCA.
• National Council for Curriculum and Assessment (NCCA). (2017). Primary mathematics curriculum. Draft specifications. Junior infant to second class. For Consultation . NCCA.
• National Council for Curriculum and Assessment (NCCA). (2020). Draft primary curriculum framework. For consultation. Primary curriculum review and development. NCCA.
• O’Shea, J., & Leavy, A. M. (2013). Teaching mathematical problem-solving from an emergent constructivist perspective: the experiences of Irish primary teachers. Journal of Mathematics Teacher Education, 16 (4), 293–318. 10.1007/s10857-013-9235-6
• Polya G. How to solve it (2nd edition) Doubleday; 1957. [ Google Scholar ]
• Russo J, Hopkins S. Teachers’ perceptions of students when observing lessons involving challenging tasks. International Journal of Science and Mathematics Education. 2019; 17 (4):759–779. doi: 10.1007/s10763-018-9888-9. [ CrossRef ] [ Google Scholar ]
• Russo J, Minas M. Student attitudes towards learning mathematics through challenging problem solving tasks: “It’s so hard-in a good way” International Electronic Journal of Elementary Education. 2020; 13 (2):215–225. [ Google Scholar ]
• Sakshaug LE, Wohlhuter KA. Journey toward teaching mathematics through problem solving. School Science and Mathematics. 2010; 110 (8):397–409. doi: 10.1111/j.1949-8594.2010.00051.x. [ CrossRef ] [ Google Scholar ]
• Shiel, G., Kavanagh, L., & Millar, D. (2014). The national assessments of english reading and mathematics: Volume 1 performance report . Educational Research Centre.
• Singer FM, Voica C. A problem-solving conceptual framework and its implications in designing problem-posing tasks. Educational Studies in Mathematics. 2013; 83 (1):9–26. doi: 10.1007/s10649-012-9422-x. [ CrossRef ] [ Google Scholar ]
• Stacey K. Teaching Mathematics through Problem Solving. Numeros. 2018; 98 :7–18. [ Google Scholar ]
• Stake RE. The art of case study research. Thousand Oaks, CA: Sage Publications; 1995. [ Google Scholar ]
• Stipek DJ, Givvin KB, Salmon JM, MacGyvers VL. Teachers’ beliefs and practices related to mathematics instruction. Teaching and Teacher Education. 2001; 17 (2):213–226. doi: 10.1016/S0742-051X(00)00052-4. [ CrossRef ] [ Google Scholar ]
• Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed). Sage.
• Sullivan P, Askew M, Cheeseman J, Clarke D, Mornane A, Roche A. Supporting teachers in structuring mathematics lessons involving challenging tasks. Journal of Mathematics Teacher Education. 2015; 18 (2):123–140. doi: 10.1007/s10857-014-9279-2. [ CrossRef ] [ Google Scholar ]
• Sullivan P, Bobis J, Downton A, Feng M, Hughes S, Livy S, McCormick M, Russo J. An instructional model to support planning and teaching student centred structured inquiry lessons. Australian Primary Mathematics Classroom. 2021; 26 (1):9–13. [ Google Scholar ]
• Sullivan, P., Clarke, D., Cheeseman, J., Mornane, A., Roche, A., Sawatzki, C., & Walker, N. (2014). Students’ willingness to engage with mathematical challenges: Implications for classroom pedagogies. In J. Anderson, M. Cavanagh, & A. Prescott (Eds.), Curriculum in focus: Research guided practice. (Proceedings of the 37th annual conference of the Mathematics Education Research Group of Australasia) (pp. 597–604). MERGA.
• Sullivan P, Clarke DM, Clarke B, O’Shea H. Exploring the relationship between task, teacher actions, and student learning. PNA. 2010; 4 (4):133–142. [ Google Scholar ]
• Suter, W. N. (2012). Introduction to educational research: A critical thinking approach (2nd ed). Sage.
• Swan M. Designing and using research instruments to describe the beliefs and practices of mathematics teachers. Research in Education. 2006; 75 (1):58–70. doi: 10.7227/RIE.75.5. [ CrossRef ] [ Google Scholar ]
• Takahashi, A. (2006). Characteristics of Japanese mathematics lessons. Paper presented at the APEC International Conference on Innovative Teaching Mathematics through Lesson Study, Tokyo, Japan, January 14–20. https://www.criced.tsukuba.ac.jp/math/sympo_2006/takahashi.pdf
• Takahashi, A. (2008). Beyond show and tell: neriage for teaching through problem-solving—ideas from Japanese problem-solving approaches for teaching mathematics. Paper presented at the 11th International Congress on Mathematics Education in Mexico (Section TSG 19: Research and Development in Problem Solving in Mathematics Education), Monteree, Mexico.
• Takahashi A. Recent trends in Japanese mathematics textbooks for elementary grades: Supporting teachers to teach mathematics through problem solving. Universal Journal of Educational Research. 2016; 4 (2):313–319. doi: 10.13189/ujer.2016.040201. [ CrossRef ] [ Google Scholar ]
• Takahashi A, Lewis C, Perry R. US lesson study network to spread teaching through problem solving. International Journal for Lesson and Learning Studies. 2013; 2 (3):237–255. doi: 10.1108/IJLLS-05-2013-0029. [ CrossRef ] [ Google Scholar ]
• Takahashi, A., & McDougal, T. (2016). Collaborative lesson research: Maximizing the impact of lesson study. ZDM: Mathematics Education, 48, 513–526.
• Thompson AG. Teachers’ conceptions of mathematics and the teaching of problem solving. In: Silver EA, editor. Teaching and learning mathematical problem solving: Multiple research perspectives. Erlbaum; 1985. pp. 281–294. [ Google Scholar ]
• Watanabe, T. (2001). Anticipating children’s thinking: A Japanese approach to instruction . National Council for Curriculum and Assessment.
• Watson, A., & Ohtani, M. (2015). Task design in mathematics education: An ICMI study 22 . Springer International.

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• DOI: 10.36948/ijfmr.2024.v06i03.21269
• Corpus ID: 270085470

Mathematics Vocabulary and Problem-solving Skills of Grade 4 Pupils in Narvacan North District

• Desiree Cabotage
• Published in International Journal For… 26 May 2024
• Mathematics, Education

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Teachers’ feedback on homework, homework-related behaviors, and academic achievement, the influence of parent education and family income on child achievement: the indirect role of parental expectations and the home environment., circles of learning: cooperation in the classroom, related papers.

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Elementary teachers’ experience of engaging with Teaching Through Problem Solving using Lesson Study

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• Published: 13 May 2022
• Volume 35 , pages 901–927, ( 2023 )

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• Mairéad Hourigan   ORCID: orcid.org/0000-0002-6895-1895 1 &
• Aisling M. Leavy   ORCID: orcid.org/0000-0002-1816-0091 1  

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For many decades, problem solving has been a focus of elementary mathematics education reforms. Despite this, in many education systems, the prevalent approach to mathematics problem solving treats it as an isolated activity instead of an integral part of teaching and learning. In this study, two mathematics teacher educators introduced 19 Irish elementary teachers to an alternative problem solving approach, namely Teaching Through Problem Solving (TTP), using Lesson Study (LS) as the professional development model. The findings suggest that the opportunity to experience TTP first-hand within their schools supported teachers in appreciating the affordances of various TTP practices. In particular, teachers reported changes in their beliefs regarding problem solving practice alongside developing problem posing knowledge. Of particular note was teachers’ contention that engaging with TTP practices through LS facilitated them to appreciate their students’ problem solving potential to the fullest extent. However, the planning implications of the TTP approach presented as a persistent barrier.

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Introduction

A fundamental goal of mathematics education is to develop students’ ability to engage in mathematical problem solving. Despite curricular emphasis internationally on problem solving, many teachers are uncertain how to harness students’ problem solving potential (Cheeseman, 2018 ). While many problem solving programmes focus on providing students with step-by-step supports through modelling, heuristics, and other structures (Polya, 1957 ), Goldenberg et al. ( 2001 ) suggest that the most effective approach to developing students’ problem solving ability is by providing them with frequent opportunities over a prolonged period to solve worthwhile open-ended problems that are challenging yet accessible to all. This viewpoint is in close alignment with reform mathematics perspectives that promote conceptual understanding, where students actively construct their knowledge and relate new ideas to prior knowledge, creating a web of connected knowledge (Hiebert, 2003 ; Lester, 2013 ; Takahashi, 2006 ; Watanabe, 2001 ).

There is consensus in the mathematics education community that problem solving should not be taught as an isolated topic focused solely on developing problem solving skills and strategies or presented as an end-of-chapter activity (Takahashi, 2006 , 2016 ; Takahashi et al., 2013 ). Instead, problem solving should be integrated across the curriculum as a fundamental part of mathematics teaching and learning (Cai & Lester, 2010 ; Takahashi, 2016 ).

A ‘Teaching Through Problem Solving’ (TTP) approach, a problem solving style of instruction that originated in elementary education in Japan, meets these criteria treating problem solving as a core practice rather than an ‘add-on’ to mathematics instruction.

Teaching Through Problem Solving (TTP)

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 understand what is required. Students then solve the problem either individually or in groups, inventing their approaches. At this stage, the teacher does not model or suggest a solution procedure. Instead, they take on the role of facilitator, providing support to students only at the right time (Hiebert, 2003 ; Lester, 2013 ; Takahashi, 2006 ). As students solve the problem, the teacher circulates, observes the range of student strategies, and identifies work that illustrates desired features. However, the problem solving lesson does not end when the students find a solution. The subsequent sharing phase, called Neriage (polishing ideas), is considered by Japanese teachers to be the heart of the lesson rather than its culmination. During Neriage, the teacher purposefully selects students to share their strategies, compares various approaches, and introduces increasingly sophisticated solution methods. Effective questioning is central to this process, alongside careful recording of the multiple solutions on the board. The teacher concludes the lesson by formalising and consolidating the lesson’s main points. This process promotes learning for all students (Hiebert, 2003 ; Stacey, 2018 ; Takahashi, 2016 ; Takahashi et al., 2013 ; Watanabe, 2001 ).

The TTP approach assumes that students develop, extend, and enrich their understandings as they confront problematic situations using existing knowledge. Therefore, TTP fosters the symbiotic relationship between conceptual understanding and problem solving, as conceptual understanding is required to solve challenging problems and make sense of new ideas by connecting them with existing knowledge. Equally, problem solving promotes conceptual understanding through the active construction of knowledge (Hiebert, 2003 ; Lambdin, 2003 ; Takahashi, 2006 ). Consequently, students simultaneously develop more profound understandings of the mathematics content while cultivating problem solving skills (Kapur, 2010 ; Stacey, 2018 ).

Relevant research affirms that teachers acknowledge the merits of this approach (Sullivan et al., 2014 ) and most students report positive experiences (Russo & Minas, 2020 ). The process is considered to make students’ thinking and learning visible (Ingram et al., 2020 ). Engagement in TTP has resulted in teachers becoming more aware of and confident in their students’ problem solving abilities and subsequently expecting more from them (Crespo & Featherstone, 2006 ; Sakshaug & Wohlhuter, 2010 ).

Demands of TTP

Adopting a TTP approach challenges pre-existing beliefs and poses additional knowledge demands for elementary teachers, both content and pedagogical (Takahashi, 2008 ).

Research has consistently reported a relationship between teacher beliefs and the instructional techniques used, with evidence of more rule-based, teacher-directed strategies used by teachers with traditional mathematics beliefs (Stipek et al., 2001 ; Swan, 2006 ; Thompson, 1985 ). These teachers tend to address problem solving separately from concept and skill development and possess a simplistic view of problem solving as translating a problem into abstract mathematical terms to solve it. Consequently, such teachers ‘are very concerned about developing skilfulness in translating (so-called) real-world problems into mathematical representations and vice versa’ (Lester, 2013 , p. 254). Early studies of problem solving practice reported direct instructional techniques where the teacher would model how to solve the problem followed by students practicing similar problems (Chapman, 2015 ; Hiebert, 2003 ; Lester, 2013 ). This naïve conception of problem solving is reflected in many textbook problems that simply require students to apply previously learned routine procedures to solve problems that are merely thinly disguised number operations (Lester, 2013 ; Singer & Voica, 2013 ). Hence, the TTP approach requires a significant shift for teachers who previously considered problem solving as an extra activity conducted after the new mathematics concepts are introduced (Lester, 2013 ; Takahashi et al., 2013 ) or whose personal experience of problem solving was confined to applying routine procedures to word problems (Sakshaug &Wohlhuter, 2010 ).

Alongside beliefs, teachers’ knowledge influences their problem solving practices. Teachers require a deep understanding of the nature of problem solving, in particular viewing problem solving as a process (Chapman, 2015 ). To be able to understand the stages problem solvers go through and appreciate what successful problem solving involves, teachers benefit from experiencing solving problems from the problem solver’s perspective (Chapman, 2015 ; Lester, 2013 ).

It is also essential that teachers understand what constitutes a worthwhile problem when selecting or posing problems (Cai, 2003 ; Chapman, 2015 ; Lester, 2013 ; O’Shea & Leavy, 2013 ). This requires an understanding that problems are ‘mathematical tasks for which the student does not have an obvious way to solve it’ (Chapman, 2015 , p. 22). Teachers need to appreciate the variety of problem characteristics that contribute to the richness of a problem, e.g. problem structures and cognitive demand (Klein & Leiken, 2020 ; O’Shea & Leavy, 2013 ). Such understandings are extensive, and rather than invest heavily in the time taken to construct their mathematics problems, teachers use pre-made textbook problems or make cosmetic changes to make cosmetic changes to these (Koichu et al., 2013 ). In TTP, due consideration must also be given to the problem characteristics that best support students in strengthening existing understandings and experiencing new learning of the target concept, process, or skill (Cai, 2003 ; Takahashi, 2008 ). Specialised content knowledge is also crucial for teachers to accurately predict and interpret various solution strategies and misconceptions/errors, to determine the validity of alternative approaches and the source of errors, to sequence student approaches, and to synthesise approaches and new learning during the TTP lesson (Ball et al., 2008 ; Cai, 2003 ; Leavy & Hourigan, 2018 ).

Teachers should also be knowledgeable regarding appropriate problem solving instruction. It is common for teachers to teach for problem solving (i.e., focusing on developing students’ problem solving skills and strategies). Teachers adopting a TTP approach engage in reform classroom practices that reflect a constructivist-oriented approach to problem solving instruction where the teacher guides students to work collaboratively to construct meaning, deciding when and how to support students without removing their autonomy (Chapman, 2015 ; Hiebert, 2003 ; Lester, 2013 ). Teachers ought to be aware of the various relevant models of problem solving, including Polya’s ( 1957 ) model that supports teaching for problem solving (Understand the problem-Devise a plan-Carry out the plan-Look back) alongside models that support TTP (e.g., Launch-Explore-Summarise) (Lester, 2013 ; Sullivan et al., 2021 ). While knowledge of heuristics and strategies may support teachers’ problem solving practices, there is consensus that teaching heuristics and strategies or teaching about problem solving does not significantly improve students’ problem solving ability. Teachers require a thorough knowledge of their students as problem solvers, for example, being aware of their abilities and factors that hinder their success, including language (Chapman, 2015 ). Knowledge of content and student, alongside content and teaching (Ball et al., 2008 ), is essential during TTP planning when predicting student approaches and errors. Such knowledge is also crucial during TTP implementation when determining the validity of alternative approaches, identifying the source of errors (Explore phase), sequencing student approaches, and synthesising the range of approaches and new learning effectively (Summarise phase) (Cai, 2003 ; Leavy & Hourigan, 2018 ).

Supports for teachers

Given the extensive demands of TTP, adopting this approach is arduous in terms of the planning time required to problem pose, predict approaches, and design questions and resources (Lester, 2013 ; Sullivan et al., 2010 ; Takahashi,  2008 ). Consequently, it is necessary to support teachers who adopt a TTP approach (Hiebert, 2003 ). Professional development must facilitate them to experience the approach themselves as learners and then provide classroom implementation opportunities that incorporate collaborative planning and reflection when trialling the approach (Watanabe, 2001 ). In Japan, a common form of professional development to promote, develop, and refine TTP implementation among teachers and test potential problems for TTP is Japanese Lesson Study (LS) (Stacey, 2018 ; Takahashi et al., 2013 ). Another valuable support is access to a repository of worthwhile problems. In Japan, government-authorised textbooks and teacher manuals provide a sequence of lessons with rich well-tested problems to introduce new concepts. They also detail alternative strategies used by students and highlight the key mathematical aspects of these strategies (Takahashi, 2016 ; Takahashi et al., 2013 ).

Teachers’ reservations about TTP

Despite the acknowledged benefits of TTP for students, some teachers report reluctance to employ TTP, identifying a range of obstacles. These include limited mathematics content knowledge or pedagogical content knowledge (Charalambous, 2008 ; Sakshaug & Wohlhuter, 2010 ) and a lack of access to resources or time to develop or modify appropriate resources (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ). Other barriers for teachers with limited experience of TTP include giving up control, struggling to support students without directing them, and a tendency to demonstrate how to solve the problem (Cheeseman, 2018 ; Crespo & Featherstone, 2006 ; Klein & Leiken, 2020 ; Takahashi et al., 2013 ). Resistance to TTP is also associated with some teachers’ perception that this approach would lead to student disengagement and hence be unsuitable for lower-performing students (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2010 ).

Problem solving practices in Irish elementary mathematics education

Within the Irish context, problem solving is a central tenet of elementary mathematics curriculum documents (Department of Education and Science (DES), 1999 ) with recommendations that problem solving should be integral to students’ mathematical learning. However, research reveals a mismatch between intended and implemented problem solving practices (Dooley et al., 2014 ; Dunphy et al., 2014 ), where classroom practices reflect a narrow approach limited to problem solving as an ‘add on’, only applied after mathematical procedures had been learned and where problems are predominantly sourced from dedicated sections of textbooks (Department of Education and Skills (DES), 2011 ; Dooley et al., 2014 ; National Council for Curriculum and Assessment (NCCA), 2016 ; O’Shea & Leavy, 2013 ). Regarding the attained curriculum, Irish students have underperformed in mathematical problem solving, relative to other skills, in national and international assessments (NCCA, 2016 ; Shiel et al., 2014 ). Consensus exists that there is scope for improvement of problem solving practices, with ongoing calls for Irish primary teachers to receive support through school-based professional development models alongside creating a repository of quality problems (DES, 2011 ; Dooley et al., 2014 ; NCCA, 2016 ).

Lesson Study (LS) as a professional development model

Reform mathematics practices, such as TTP, challenge many elementary teachers’ beliefs, knowledge, practices, and cultural norms, particularly if they have not experienced the approach themselves as learners. To support teachers in enacting reform approaches, they require opportunities to engage in extended and targeted professional development involving collaborative and practice-centred experiences (Dudley et al., 2019 ; Murata et al., 2012 ; Takahashi et al., 2013 ). Lesson Study (LS) possesses the characteristics of effective professional development as it embeds ‘…teachers’ learning in their everyday work…increasing the likelihood that their learning will be meaningful’ (Fernandez et al., 2003 , p. 171).

In Japan, LS was developed in the 1980s to support teachers to use more student-centred practices. LS is a school-based, collaborative, reflective, iterative, and research-based form of professional development (Dudley et al., 2019 ; Murata et al., 2012 ). In Japan, LS is an integral part of teaching and is typically conducted as part of a school-wide project focused on addressing an identified teaching–learning challenge (Takahashi & McDougal, 2016 ). It involves a group of qualified teachers, generally within a single school, working together as part of a LS group to examine and better understand effective teaching practices. Within the four phases of the LS cycle, the LS group works collaboratively to study and plan a research lesson that addresses a pre-established goal before implementing (teach) and reflecting (observe, analyse and revise) on the impact of the lesson activities on students’ learning.

LS has become an increasingly popular professional development model outside of Japan in the last two decades. In these educational contexts, it is necessary to find a balance between fidelity to LS as originally envisaged and developing a LS approach that fits the cultural context of a country’s education system (Takahashi & McDougal, 2016 ).

Relevant research examining the impact of LS on qualified primary mathematics teachers reports many benefits. Several studies reveal that teachers demonstrated transformed beliefs regarding effective pedagogy and increased self-efficacy in their use due to engaging in LS (Cajkler et al., 2015 ; Dudley et al., 2019 ; Fernandez, 2005 ; Gutierez, 2016 ). Enhancements in participating teachers’ knowledge have also been reported (Cajkler et al., 2015 ; Dudley et al., 2019 ; Fernandez, 2005 ; Gutierez, 2016 ; Murata et al., 2012 ). Other gains recounted include improvements in practice with a greater focus on students (Cajkler et al., 2015 ; Dudley et al., 2019 ; Flanagan, 2021 ).

Context of this study

A cluster of urban schools, coordinated by their local Education Centre, engaged in an initiative to enhance teachers’ mathematics problem solving practices. The co-ordinator of the initiative approached the researchers, both mathematics teacher educators (MTEs), seeking a relevant professional development opportunity. Aware of the challenges of problem solving practice within the Irish context, the MTEs proposed an alternative perspective on problem solving: the Teaching Through Problem Solving (TTP) approach. Given Cai’s ( 2003 ) recommendation that teachers can best learn to teach through problem solving by teaching and reflecting as opposed to taking more courses, the MTEs identified LS as the best fit in terms of a supportive professional development model, as it is collaborative, experiential, and school-based (Dudley et al., 2019 ; Murata et al., 2012 ; Takahashi et al., 2013 ). Consequently, LS would promote teachers to work collaboratively to understand the TTP approach, plan TTP practices for their educational context, observe what it looks like in practice, and assess the impact on their students’ thinking (Takahashi et al., 2013 ). In particular, the MTEs believed that the LS phases and practices would naturally support TTP structures, emphasizing task selection and anticipating students’ solutions. Given Lester’s ( 2013 ) assertion that each problem solving experience a teacher engages in can potentially alter their knowledge for teaching problem solving, the MTEs sought to explore teachers’ perceptions of the impact of engaging with TTP through LS on their beliefs regarding problem solving and their knowledge for teaching problem solving.

Research questions

This paper examines two research questions:

Research question 1: What are elementary teachers’ reported problem solving practices prior to engaging in LS?

Research question 2: What are elementary teachers’ perceptions of what they learned from engaging with TTP through LS?

Methodology

Participants.

The MTEs worked with 19 elementary teachers (16 female, three male) from eight urban schools. Schools were paired to create four LS groups on the basis of the grade taught by participating class teachers, e.g. Grade 3 teacher from school 1 paired with Grade 4 teacher from school 2. Each LS group generally consisted of 4–5 teachers, with a minimum of two teachers from each school, along with the two MTEs. For most teachers, LS and TTP were new practices being implemented concurrently. However, given the acknowledged overlap between the features of the TTP and LS approaches, for example, the focus on problem posing and predicting student strategies, the researchers were confident that the content and structure were compatible. Also, in Japan, LS is commonly used to promote TTP implementation among teachers (Stacey, 2018 ; Takahashi et al., 2013 ).

All ethical obligations were adhered to throughout the research process, and the study received ethical approval from the researchers’ institutional board. Of the 19 participating LS teachers invited to partake in the research study, 16 provided informed consent to use their data for research purposes.

Over eight weeks, the MTEs worked with teachers, guiding each LS group through the four LS phases involving study, design, implementation, and reflection of a research lesson that focused on TTP while assuming the role of ‘knowledgeable others’ (Dudley et al., 2019 ; Hourigan & Leavy, 2021 ; Takahashi & McDougal, 2016 ). An overview of the timeline and summary of each LS phase is presented in Table 1 .

LS phase 1: Study

This initial study phase involved a one-day workshop. The process and benefits of LS as a school-based form of professional development were discussed in the morning session and the afternoon component was spent focusing on the characteristics of TTP. Teachers experienced the TPP approach first-hand by engaging in the various lesson stages. For example, they solved a problem (growing pattern problem) themselves in pairs and shared their strategies. They also predicted children’s approaches to the problem and possible misconceptions and watched the video cases of TTP classroom practice for this problem. Particular focus was placed on the importance of problem selection and prediction of student strategies before the lesson implementation and the Neriage stage of the lesson. Teachers also discussed readings related to LS practices (e.g. Lewis & Tsuchida, 1998 ) and TTP (e.g., Takahashi, 2008 ). At the end of the workshop, members of each LS group were asked to communicate among themselves and the MTEs, before the planning phase, to decide the specific mathematics focus of their LS group’s TTP lesson (Table 1 ).

LS phase 2: Planning

The planning phase was four weeks in duration and included two 1½ hour face-to-face planning sessions (i.e. planning meetings 1 and 2) between the MTEs and each LS group (Table 1 ). Meetings took place in one of the LS group’s schools. At the start of the first planning meeting, time was dedicated to Takahashi’s ( 2008 ) work focusing on the importance of problem selection and prediction of student strategies to plan the Neriage stage of the TTP lesson. The research lesson plan structure was also introduced. Ertle et al.’s ( 2001 ) four column lesson plan template was used. It was considered particularly compatible with the TTP approach, given the explicit attention to expected student response and the teacher’s response to student activity/response.

The planning then moved onto the content focus of each LS group’s TTP research lesson. LS groups selected TTP research lessons focusing on number (group A), growing patterns (group B), money (group C), and 3D shapes (group D). Across the planning phase, teachers invested substantial time extensively discussing the TTP lesson goals in terms of target mathematics content, developing or modifying a problem to address these goals, and exploring considerations for the various lesson stages. Drawing on Takahashi’s ( 2008 ) article, it was re-emphasised that no strategies would be explicitly taught before students engaged with the problem. While one LS group modified an existing problem (group B) (Hourigan & Leavy, 2015 ), the other three LS groups posed an original problem. To promote optimum teacher readiness to lead the Neriage stage, each LS group was encouraged to solve the problem themselves in various ways considering possible student strategies and their level of mathematical complexity, thus identifying the most appropriate sequence of sharing solutions.

LS phase 3: Implementation

The implementation phase involved one teacher in each LS group teaching the research lesson (teach 1) in their school. The remaining group members and MTEs observed and recorded students’ responses. Each LS group and the MTEs met immediately for a post-lesson discussion to evaluate the research lesson. The MTEs presented teachers with a series of focus questions: What were your observations of student learning? Were the goals of the lesson achieved? Did the problem support students in developing the appropriate understandings? Were there any strategies/errors that we had not predicted? How did the Neriage stage work? What aspects of the lesson plan should be reconsidered based on this evidence? Where appropriate, the MTEs drew teachers’ attention to particular lesson aspects they had not noticed. Subsequently, each LS group revised their research lesson in response to the observations, reflections, and discussion. The revised lesson was retaught 7–10 days later by a second group member from the paired LS group school (teach 2) (Table 1 ). The post-lesson discussion for teach 2 focused mainly on the impact of changes made after the first implementation on student learning, differences between the two classes, and further changes to the lesson.

LS phase 4: Reflection

While reflection occurred after both lesson implementations, the final reflection involved all teachers from the eight schools coming together for a half-day meeting in the local Education Centre to share their research lessons, experiences, and learning (Table 1 ). Each LS group made a presentation, identifying their research lesson’s content focus and sequence of activity. Artefacts (research lesson plan, materials, student work samples, photos) were used to support observations, reflections, and lesson modifications. During this meeting, teachers also reflected privately and in groups on their initial thoughts and experience of both LS and TTP, the benefits of participation, the challenges they faced, and they provided suggestions for future practice.

Data collection

The study was a collective case study (Stake, 1995 ). Each LS group constituted a case; thus, the analysis was structured around four cases. Data collection was closely aligned with and ran concurrent to the LS process. Table 2 details the links between the LS phases and the data collection process.

The principal data sources (Table 2 ) included both MTEs’ fieldnotes (phase (P) 1–4), and reflections (P1–4), alongside email correspondence (P1–4), individual teacher reflections (P1, 2, 4) (see reflection tasks in Table 3 ), and LS documentation including various drafts of lesson plans (P2–4) and group presentations (P4). Fieldnotes refer to all notes taken by MTEs when working with the LS groups, for example, during the study session, planning meetings, lesson implementations, post-lesson discussions, and the final reflection session.

The researchers were aware of the limitations of self-report data and the potential mismatch between one’s perceptions and reality. Furthermore, data in the form of opinions, attitudes, and beliefs may contain a certain degree of bias. However, this paper intentionally focuses solely on the teachers’ perceived learning in order to represent their ‘lived experience’ of TTP. Despite this, measures were taken to assure the trustworthiness and rigour of this qualitative study. The researchers engaged with the study over a prolonged period and collected data for each case (LS group) at every LS phase (Table 2 ). All transcripts reflected verbatim accounts of participants’ opinions and reflections. At regular intervals during the study, research meetings interrogated the researchers’ understandings, comparing participating teachers’ observations and reflections to promote meaning-making (Creswell, 2009 ; Suter, 2012 ).

Data analysis

The MTEs’ role as participant researchers was considered a strength of the research given that they possessed unique insights into the research context. A grounded theory approach was adopted, where the theory emerges from the data analysis process rather than starting with a theory to be confirmed or refuted (Glaser, 1978 ; Strauss & Corbin, 1998 ). Data were examined focusing on evidence of participants’ problem solving practices prior to LS and their perceptions of their learning as a result of engaging with TTP through LS. A systematic process of data analysis was adopted. Initially, raw data were organised into natural units of related data under various codes, e.g. resistance, traditional approach, ignorance, language, planning, fear of student response, relevance, and underestimation. Through successive examinations of the relationship between existing units, codes were amalgamated (Creswell, 2009 ). Progressive drafts resulted in the firming up of several themes. Triangulation was used to establish consistency across multiple data sources. While the first theme, Vast divide between prevalent problem solving practices and TTP , addresses research question 1, it is considered an overarching theme, given the impact of teachers’ established problem solving understandings and practices on their receptiveness to and experience of TTP. The remaining five themes ( Seeing is believing : the value of practice centred experiences ; A gained appreciation of the relevance and value of TTP practices ; Enhanced problem posing understandings ; Awakening to students’ problem solving potential ; and Reservations regarding TTP) represent a generalised model of teachers’ perceived learning due to engaging with TTP through LS, thus addressing research question 2. Although one of the researchers was responsible for the initial coding, both researchers met regularly during the analysis to discuss and interrogate the established codes and to agree on themes. This process served to counteract personal bias (Suter, 2012 ).

As teacher reflections were anonymised, it was not possible to track teachers across LS phases. Consequently, teacher reflection data are labelled as phase and instrument only. For example, ‘P2, teacher reflection’ communicates that the data were collected during LS phase 2 through teacher reflection. However, the remaining data are labelled according to phase, instrument, and source, e.g. ‘P3, fieldnotes: group B’. While phase 4 data reflect teachers’ perceptions after engaging fully with the TTP approach, data from the earlier phases reflect teachers’ evolving perceptions at a particular point in their unfolding TTP experience.

Discussion of findings

The findings draw on the analysis of the data collected across the LS phases and address the research questions. Within the confines of this paper, illustrative quotes are presented to provide insights into each theme. An additional layer of analysis was completed to ensure a balanced representation of teachers’ views in reporting findings. This process confirmed that the findings represent the views of teachers across LS groups, for example, within the first theme presented ( Vast divide ), the eight quotes used came from eight different teacher reflections. Equally, the six fieldnote excerpts selected represent six different teachers’ views across the four LS groups. Furthermore, in the second theme ( Seeing is believing ), the five quotes presented were sourced from five different participating teachers’ reflections and the six fieldnote excerpts included are from six different teachers across the four LS groups. Subsequent examination of the perceptions of those teachers not included in the reporting of findings confirmed that their perspectives were represented within the quotes used. Hence, the researchers are confident that the findings represent the views of teachers across all LS groups. For each theme, sources of evidence that informed the presented conclusions will be outlined.

Vast divide between prevalent problem solving practices and TTP

This overarching theme addresses the research question ‘What were elementary teachers’ reported problem solving practices prior to engaging in LS?’.

At the start of the initiative, within the study session (fieldnotes), all teachers identified mathematics problem solving as a problem of practice. The desire to develop problem solving practices was also apparent in some teachers’ reflections (phase 1 (P1), N  = 8):

I am anxious about it. Problem solving is an area of great difficulty throughout our school (P1, teacher reflection).

During both study and planning phase discussions, across all LS groups, teachers’ reports suggested the almost exclusive use of a teaching for problem solving approach, with no awareness of the Teaching Through Problem Solving (TTP) approach; a finding also evidenced in both teacher reflections (P1, N  = 7) and email correspondence:

Unfamiliar, not what I am used to. I have no experience of this kind of problem solving. This new approach is the reverse way to what I have used for problem solving (P1, teacher reflection) Being introduced to new methods of teaching problem solving and trying different approaches is both exciting and challenging (P1: email correspondence)

Teachers’ descriptions of their problem solving classroom practices in both teacher reflections (P1, N  = 8) and study session discussions (fieldnotes) suggested a naïve conception of problem solving, using heuristics such as the ‘RUDE (read, underline, draw a picture, estimate) strategy’ (P1, fieldnotes) to support students in decoding and solving the problem:

In general, the problem solving approach described by teachers is textbook-led, where concepts are taught context free first and the problems at the end of the chapter are completed afterwards (P1, reflection: MTE2)

This approach was confirmed as widespread across all LS groups within the planning meetings (fieldnotes).

In terms of problem solving instruction, a teacher-directed approach was reported by some teachers within teacher reflections (P1, N  = 5), where the teacher focused on a particular strategy and modelled its use by solving the problem:

I tend to introduce the problem, ensure everyone understands the language and what is being asked. I discuss the various strategies that children could use to solve the problem. Sometimes I demonstrate the approach. Then children practice similar problems … (P1: Teacher reflection)

However, it was evident within the planning meetings, that this traditional approach to problem solving was prevalent among the teachers in all LS groups. During the study session (field notes and teacher reflections (P1, N  = 7)), there was a sense that problem solving was an add-on as opposed to an integral part of mathematics teaching and learning. Again, within the planning meetings, discussions across all four LS groups verified this:

Challenge: Time to focus on problems not just computation (P1: Teacher reflection). From our discussions with the various LS groups’ first planning meeting, text-based teaching seems to be resulting in many teachers teaching concepts context-free initially and then matching the concept with the relevant problems afterwards (P2, reflection: MTE1)

However, while phase 1 teacher reflections suggested that a small number of participating teachers ( N  = 4) possessed broader problem solving understandings, subsequently during the planning meetings, there was ample evidence (field notes) of problem-posing knowledge and the use of constructivist-oriented approaches that would support the TTP approach among some participating teachers in each of the LS groups:

Challenge: Spend more time on meaningful problems and give them opportunities and time to engage in activities, rather than go too soon into tricks, rhymes etc (P1, Teacher reflection). The class are already used to sharing strategies and explaining where they went wrong (P2, fieldnotes, Group B) Teacher: The problem needs to have multiple entry points (P2, fieldnotes: Group C)

While a few teachers reported problem posing practices, in most cases, this consisted of cosmetic adjustments to textbook problems. Overall, despite evidence of some promising practices, the data evidenced predominantly traditional problem solving views and practices among participating teachers, with potential for further broadening of various aspects of their knowledge for teaching problem solving including what constitutes a worthwhile problem, the role of problem posing within problem solving, and problem solving instruction. Within phase 1 teacher reflections, when reporting ‘challenges’ to problem solving practices (Table 3 ), a small number of responses ( N  = 3) supported these conclusions:

Differences in teachers’ knowledge (P1: Teacher reflection). Need to challenge current classroom practices (P1: Teacher reflection).

However, from the outset, all participating teachers consistently demonstrated robust knowledge of their students as problem solvers, evidenced in phase 1 teacher reflections ( N  = 10) and planning meeting discussions (P2, fieldnotes). However, in these early phases, teachers generally portrayed a deficit view, focusing almost exclusively on the various challenges impacting their students’ problem solving abilities. While all teachers agreed that the language of problems was inhibiting student engagement, other common barriers reported included student motivation and perseverance:

They often have difficulties accessing the problem – they don’t know what it is asking them (P2, fieldnotes: Group C) Sourcing problems that are relevant to their lives. I need to change every problem to reference soccer so the children are interested (P1: teacher reflection) Our children deal poorly with struggle and are slow to consider alternative strategies (P2, fieldnotes: Group D)

Despite showcasing a strong awareness of their students’ problem solving difficulties, teachers initially demonstrated a lack of appreciation of the benefits accrued from predicting students’ approaches and misconceptions relating to problem solving. While it came to the researchers’ attention during the study phase, its prevalence became apparent during the initial planning meeting, as its necessity and purpose was raised in three of the LS groups:

What are the benefits of predicting the children’s responses? (P1, fieldnotes). I don’t think we can predict- we will have to wait and see (P2, fieldnotes: Group A).

This finding evidences teachers’ relatively limited knowledge for teaching problem solving, given that this practice is fundamental to TTP and constructivist-oriented approaches to problem solving instruction.

Perceived impacts of engaging with TTP through LS

In response to the research question ‘What are elementary teachers’ perceptions of what they learned from engaging with TTP through LS?’, thematic data analysis identified 5 predominant themes, namely, Seeing is believing : the value of practice centred experiences ; A gained appreciation of the relevance and value of TTP practices ; Enhanced problem posing understandings ; Awakening to students’ problem solving potential ; and Reservations regarding TTP.

Seeing is believing: the value of practice centred experiences

Teachers engaged with TTP during the study phase as both learners and teachers when solving the problem. They were also involved in predicting and analysing student responses when viewing the video cases, and engaged in extensive reading, discussion, and planning for their selected TTP problem within the planning phase. Nevertheless, teachers reported reservations about the relevance of TTP for their context within both phase 2 teacher reflections ( N  = 5) as well as within the planning meeting discourse of all LS groups. Teachers’ keen awareness of their students’ problem solving challenges, coupled with the vast divide between the nature of their prior problem solving practices and the TTP approach, resulted in teachers communicating concern regarding students’ possible reaction during the planning phase:

I am worried about the problem. I am concerned that if the problem is too complex the children won’t respond to it (P2, fieldnotes: Group B) The fear that the children will not understand the lesson objective. Will they engage? (P2, Teacher reflection)

Acknowledging their apprehension regarding students’ reactions to TTP, from the outset, all participating teachers communicated a willingness to trial TTP practices:

Exciting to be part of. Eager to see how it will pan out and the learning that will be taken from it (P1, teacher reflection) They should be ‘let off’ (P2, fieldnotes: Group A).

It was only within the implementation phase, when teachers received the opportunity to meaningfully observe the TTP approach in their everyday work context, with their students, that they explicitly demonstrated an appreciation for the value of TTP practices. It was evident from teacher commentary across all LS groups’ post-lesson discussions (fieldnotes) as well as in teacher reflections (P4, N  = 10) that observing first-hand the high levels of student engagement alongside students’ capacity to engage in desirable problem solving strategies and demonstrate sought-after dispositions had affected this change:

Class teacher: They engaged the whole time because it was interesting to them. The problem is core in terms of motivation. It determines their willingness to persevere. Otherwise, it won’t work whether they have the skills or not (P3, fieldnotes: Group C) LS group member: The problem context worked really well. The children were all eager and persevered. It facilitated all to enter at their own level, coming up with ideas and using their prior knowledge to solve the problem. Working in pairs and the concrete materials were very supportive. It’s something I’d never have done before (P3, fieldnotes: Group A)

Although all teachers showcased robust knowledge of their students’ problem solving abilities prior to engaging in TTP, albeit with a tendency to focus on their difficulties and factors that inhibited them, teachers’ contributions during post-lesson discussions (fieldnotes) alongside teacher reflections (P4, N  = 9) indicate that observing TTP in action supported them in developing an appreciation of value of the respective TTP practices, particularly the role of prediction and observation of students’ strategies/misconceptions in making the students’ thinking more visible:

You see the students through the process (P3, fieldnotes: Group C) It’s rare we have time to think, to break the problem down, to watch and understand children’s ways of thinking/solving. It’s really beneficial to get a chance to re-evaluate the teaching methods, to edit the lesson, to re-teach (P4, teacher reflection)

Analysis of the range of data sources across the phases suggests that it was the opportunity to experience TTP in practice in their classrooms that provided the ‘proof of concept’:

I thought it wasn’t realistic but bringing it down to your own classroom it is relevant (P4, teacher reflection).

Hence from the teachers’ perspective, they witnessed the affordances of TTP practices in the implementation phase of the LS process.

A gained appreciation of the relevance and value of TTP practices

While during the early LS phases, teachers’ reporting suggested a view of problem solving as teaching to problem solve, data from both fieldnotes (phases 3 and 4) and teacher reflections (phase 4) demonstrate that all teachers broadened their understanding of problem solving as a result of engaging with TTP:

Interesting to turn lessons on their head and give students the chance to think, plan and come up with possible strategies and solutions (P4, Teacher reflection)

On witnessing the affordances of TTP first-hand in their own classrooms, within both teacher reflections (P4, N  = 12) and LS group presentations, the teachers consistently reported valuing these new practices:

I just thought the whole way of teaching was a good way, an effective way of teaching. Sharing and exploring more than one way of solving is vital (P4, teacher reflection) There is a place for it in the classroom. I will use aspects of it going forward (P4, fieldnotes: Group C)

In fact, teachers’ support for this problem solving approach was apparent in phase 3 during the initial post-lesson discussions. It was particularly notable when a visitor outside of the LS group who observed teach 1 challenged the approach, recommending the explicit teaching of strategies prior to engagement. A LS group member’s reply evidenced the group’s belief that TTP naturally exposes students to the relevant learning: ‘Sharing and questioning will allow students to learn more efficient strategies [other LS group members nodding in agreement]’ (P3, fieldnotes; Group A).

In turn, within phase 4 teacher reflections, teachers consistently acknowledged that engaging with TTP through LS had challenged their understandings about what constitutes effective problem solving instruction ( N  = 12). In both teacher reflections (P4, N  = 14) and all LS group presentations, teachers reported an increased appreciation of the benefits of adopting a constructivist-oriented approach to problem solving instruction. Equally for some, this was accompanied by an acknowledgement of a heightened awareness of the limitations of their previous practice :

Really made me re-think problem solving lesson structures. I tend to spoon-feed them …over-scaffold, a lot of teacher talk. … I need to find a balance… (P4, teacher reflection) Less is more, one problem can be the basis for an entire lesson (P4, teacher reflection)

What was unexpected, was that some teachers (P4, N  = 8) reported that engaging with TTP through LS resulted in them developing an increased appreciation of the value of problem solving and the need for more regular opportunities for students to engage in problem solving:

I’ve come to realise that problem solving is critical and it should be focused on more often. I feel that with regular exposure to problems they’ll come to love being problem solvers (P4, teacher reflection)

Enhanced problem posing understandings

In the early phases of LS, few teachers demonstrated familiarity with problem characteristics (P2 teacher reflection, N  = 5). However, there was growth in teachers’ understandings of what constitutes a worthwhile problem and its role within TTP within all LS groups’ post-lesson discussions and presentations (fieldnotes) and teacher reflections (P4, N  = 10):

I have a deepened understanding of how to evaluate a problem (P2, teacher reflection) It’s essential to find or create a good problem with multiple strategies and/or solutions as a springboard for a topic. It has to be relevant and interesting for the kids (P4, teacher reflection)

As early as the planning phase, a small group of teachers’ reflections ( N  = 2) suggested an understanding that problem posing is an important aspect of problem solving that merits significant attention:

It was extremely helpful to problem solve the problem (P2, teacher reflection)

However, during subsequent phases, this realisation became more mainstream, evident within all LS groups’ post-lesson discussions and presentations (fieldnotes) and teacher reflections (P4, N  = 12):

During the first planning meeting, I was surprised and a bit anxious that we would never get to having created a problem. In hindsight, this was time well spent as the problem was crucial (P4, teacher reflection) I learned the problem is key. We don’t spend enough time picking the problem (P4, fieldnotes: Group C).

Alongside this, in all LS groups’ dialogues during the post-lesson discussion and presentations (fieldnotes) and teacher reflections (P4, N  = 15), teachers consistently demonstrated an enhanced awareness of the interdependence between the quality of the problem and students’ problem solving behaviours:

Better perseverance if the problem is of interest to them (P4, teacher reflection) It was an eye-opener to me, relevance is crucial, when the problem context is relevant to them, they are motivated to engage and can solve problems at an appropriate level…They all wanted to present (P3, fieldnotes: Group C)

The findings suggest that engaging with TTP through LS facilitated participating teachers to develop an enhanced understanding of the importance of problem posing and in identifying the features of a good mathematics problem, thus developing their future problem posing capacity. In essence, the opportunity to observe the TTP practices in their classrooms stimulated an enhanced appreciation for the value of meticulous attention to detail in TTP planning.

Awakening to students’ problem solving potential

In the final LS phases, teachers consistently reported that engaging with TTP through LS provided the opportunity to see the students through the process , thus supporting them in examining their students’ capabilities more closely. Across post-lesson discussions and presentations (fieldnotes) and teacher reflections (P4, N  = 14), teachers acknowledged that engagement in core TTP practices, including problem posing, prediction of students’ strategies during planning, and careful observation of approaches during the implementation phase, facilitated them to uncover the true extent of their students’ problem solving abilities, heightening their awareness of students’ proficiency in using a range of approaches:

Class teacher: While they took a while to warm up, I am most happy that they failed, tried again and succeeded. They all participated. Some found a pattern, others used trial and error. Others worked backward- opening the cube in different ways. They said afterward ‘That was the best maths class ever’ (P3, fieldnotes: Group D) I was surprised with what they could do. I have learned the importance of not teaching strategies first. I need to pull back and let the children solve the problems their own way and leave discussing strategies to the end (P4, teacher reflection)

In three LS groups, class teachers acknowledged in the post-lesson discussion (fieldnotes) that engaging with TTP had resulted in them realising their previous underestimation of [some or all] of their students’ problem solving abilities . Teacher reflections (P4, N  = 8) and LS group presentations (fieldnotes) also acknowledged this reality:

I underestimated my kids, which is awful. The children surprised me with the way they approached the problem. In the future I need to focus on what they can do as much as what might hinder them…they are more able than we may think (P4, reflection)

In all LS groups, teachers reported that their heightened appreciation of students’ problem solving capacities promoted them to use a more constructivist-orientated approach in the future:

I learned to trust the students to problem solve, less scaffolding. Children can be let off to explore without so much teacher intervention (P3, fieldnotes: Group D)

Some teachers ( N  = 3) also acknowledged the affective benefits of TTP on students:

I know the students enjoyed sharing their different strategies…it was great for their confidence (P4, teacher reflection)

Interestingly, in contrast with teachers’ initial reservations, their experiential and school-based participation in TTP through LS resulted in a lessening of concern regarding the suitability of TTP practices for their students. Hence, this practice-based model supported teachers in appreciating the full extent of their students’ capacities as problem solvers.

Reservations regarding TTP

When introduced to the concept of TTP in the study session, one teacher quickly addressed the time implications:

It is unrealistic in the everyday classroom environment. Time is the issue. We don’t have 2 hours to prep a problem geared at the various needs (P1, fieldnotes)

Subsequently, across the initiative, during both planning meetings, the reflection session and individual reflections (P4, N  = 14), acknowledgements of the affordances of TTP practices were accompanied by questioning of its sustainability due to the excessive planning commitment involved:

It would be hard to maintain this level of planning in advance of the lesson required to ensure a successful outcome (P4, teacher reflection)

Given the extensive time dedicated to problem posing, solving, prediction, and design of questions as well as selection or creation of materials both during and between planning meetings, there was agreement in the reflection session (fieldnotes) and in teacher reflections (P4, N  = 10) that while TTP practices were valuable, in the absence of suitable support materials for teachers, adjustments were essential to promote implementation:

There is definitely a role for TTP in the classroom, however the level of planning involved would have to be reduced to make it feasible (P4, teacher reflection) The TTP approach is very effective but the level of planning involved is unrealistic with an already overcrowded curriculum. However, elements of it can be used within the classroom (P4, teacher reflection)

A few teachers ( N  = 3) had hesitations beyond the time demands, believing the success of TTP is contingent on ‘a number of criteria…’ (P4, teacher reflection):

A whole-school approach is needed, it should be taught from junior infants (P4, teacher reflection) I still have worries about TTP. We found it difficult to decide a topic initially. It lends itself to certain areas. It worked well for shape and space (P4, teacher reflection)

Conclusions

The reported problem solving practice reflects those portrayed in the literature (NCCA, 2016 ; O’Shea & Leavy, 2013 ) and could be aptly described as ‘pendulum swings between emphases on basic skills and problem solving’ (Lesh & Zawojewski, 2007 in Takahashi et al., 2013 , p. 239). Teachers’ accounts depicted problem solving as an ‘add on’ occurring on an ad hoc basis after concepts were taught (Dooley et al., 2014 ; Takahashi et al., 2013 ), suggesting a simplistic view of problem solving (Singer & Voica, 2013 ; Swan, 2006 ). Hence, in reality there was a vast divide between teachers’ problem solving practices and TTP. Alongside traditional beliefs and problem solving practices (Stipek et al., 2001 ; Swan, 2006 ; Thompson, 1985 ), many teachers demonstrated limited insight regarding what constitutes a worthwhile problem (Klein & Lieken, 2020 ) or the critical role of problem posing in problem solving (Cai, 2003 ; Takahashi, 2008 ; Watson & Ohtani, 2015 ). Teachers’ reports suggested most were not actively problem posing, with reported practices limited to cosmetic changes to the problem context (Koichu et al., 2013 ). Equally, teachers demonstrated a lack of awareness of alternative approaches to teaching for problem solving (Chapman, 2015 ) alongside limited appreciation among most of the affordances of a more child-centred approach to problem solving instruction (Hiebert, 2003 ; Lester, 2013 ; Swan, 2006 ). Conversely, there was evidence that some teachers held relevant problem posing knowledge and utilised practices compatible with the TTP approach.

All teachers displayed relatively strong understandings of their students as problem solvers from the outset; however, they initially focused almost exclusively on factors impacting students’ limited problem solving capacity (Chapman, 2015 ). Teachers’ perceptions of their students’ problem solving abilities alongside the vast divide between teachers’ problem solving practice and the proposed TTP approach resulted in teachers being initially concerned regarding students’ response to TTP. This finding supports studies that reported resistance by teachers to the use of challenging tasks due to fears that students would not be able to manage (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2010 ). Equally, teachers communicated disquiet from the study phase regarding the time investment required to adopt the TTP approach, a finding common in similar studies (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ). Hence, the transition to TTP was uneasy for most teachers, given the significant shift it represented in terms of moving beyond a teaching to problem solve approach alongside the range of teacher demands (Takahashi et al., 2013 ).

Nevertheless, despite initial reservations, all teachers reported that engagement with TTP through LS affected their problem solving beliefs and understandings. What was particularly notable was that they reported an awakening to students’ problem solving potential . During LS’s implementation and reflection stages, all teachers acknowledged that seeing was believing concerning the benefits of TTP for their students (Kapur, 2010 ; Stacey, 2018 ). In particular, they recognised students’ positive response (Russo & Minas, 2020 ) enacted in high levels of engagement, perseverance in finding a solution, and the utilisation of a range of different strategies. These behaviours were in stark contrast to teachers’ reports in the study phase. Teachers acknowledged that students had more potential to solve problems autonomously than they initially envisaged. This finding supports previous studies where teachers reported that allowing students to engage with challenging tasks independently made students’ thinking more visible (Crespo & Featherstone, 2006 ; Ingram et al., 2020 ; Sakshand & Wohluter, 2010 ). It also reflects Sakshaug and Wohlhuter’s ( 2010 ) findings of teachers’ tendency to underestimate students’ potential to solve problems. Interestingly, at the end of LS, concern regarding the appropriateness of the TTP approach for students was no longer cited by teachers. This finding contrasts with previous studies that report teacher resistance due to fears that students will become disengaged due to the unsuitability of the approach (challenging tasks) for lower-performing students (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ). Hence, engaging with TTP through LS supported teachers in developing an appreciation of their students’ potential as problem solvers.

Teachers reported enhanced problem posing understandings, consisting of newfound awareness of the connections between the quality of the problem, the approach to problem solving instruction, and student response (Chapman, 2015 ; Cai, 2003 ; Sullivan et al., 2015 ; Takahashi, 2008 ). They acknowledged that they had learned the importance of the problem in determining the quality of learning and affecting student engagement, motivation and perseverance, and willingness to share strategies (Cai, 2003 ; Watson & Oktani, 2015 ). These findings reflect previous research reporting that engagement in LS facilitated teachers to enhance their teacher knowledge (Cajkler et al., 2015 ; Dudley et al., 2019 ; Gutierez, 2016 ).

While all teachers acknowledged the benefits of the TTP approach for students (Cai & Lester, 2010 ; Sullivan et al., 2014 ; Takahashi, 2016 ), the majority confirmed their perception of the relevance and value of various TTP practices (Hiebert, 2003 ; Lambdin, 2003 ; Takahashi, 2006 ). They referenced the benefits of giving more attention to the problem, allowing students the opportunity to independently solve, and promoting the sharing of strategies and pledged to incorporate these in their problem solving practices going forward. Many verified that the experience had triggered them to question their previous problem solving beliefs and practices (Chapman, 2015 ; Lester, 2013 ; Takahashi et al., 2013 ). This study supports previous research reporting that LS challenged teachers’ beliefs regarding the characteristics of effective pedagogy (Cajkler et al., 2015 ; Dudley et al., 2019 ; Fernandez, 2005 ; Gutierez, 2016 ). However, teachers communicated reservations regarding TTP , refraining from committing to TTP in its entirety, highlighting that the time commitment required for successful implementation on an ongoing basis was unrealistic. Therefore, teachers’ issues with what they perceived to be the excessive resource implications of TTP practices remained constant across the initiative. This finding supports previous studies that report teachers were resistant to engaging their students with ‘challenging tasks’ provided by researchers due to the time commitment required to plan adequately (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ).

Unlike previous studies, teachers in this study did not perceive weak mathematics content or pedagogical content knowledge as a barrier to implementing TTP (Charalambous, 2008 ; Sakshaug & Wohlhuter, 2010 ). However, it should be noted that the collaborative nature of LS may have hidden the knowledge demands for an individual teacher working alone when engaging in the ‘Anticipate’ element of TTP particularly in the absence of appropriate supports such as a bank of suitable problems.

The findings suggest that LS played a crucial role in promoting reported changes, serving both as a supportive professional development model (Stacey, 2018 ; Takahashi et al., 2013 ) and as a catalyst, providing teachers with the opportunity to engage in a collaborative, practice-centred experience over an extended period (Dudley et al., 2019 ; Watanabe, 2001 ). The various features of the LS process provided teachers with opportunities to engage with, interrogate, and reflect upon key TTP practices. Reported developments in understandings and beliefs were closely tied to meaningful opportunities to witness first-hand the affordances of the TTP approach in their classrooms with their students (Dudley et al., 2019 ; Fernandez et al., 2003 ; Takahashi et al., 2013 ). We suggest that the use of traditional ‘one-off’ professional development models to introduce TTP, combined with the lack of support during the implementation phase, would most likely result in teachers maintaining their initial views about the unsuitability of TTP practices for their students.

In terms of study limitations, given that all data were collected during the LS phases, the findings do not reflect the impact on teachers’ problem solving classroom practice in the medium to long term. Equally, while acknowledging the limitations of self-report data, there was no sense that the teachers were trying to please the MTEs, as they were forthright when invited to identify issues. Also, all data collected through teacher reflection was anonymous. The relatively small number of participating teachers means that the findings are not generalisable. However, they do add weight to the body of relevant research. This study also contributes to the field as it documents potential challenges associated with implementing TTP for the first time. It also suggests that despite TTP being at odds with their problem solving practice and arduous, the opportunity to experience the impact of the TTP approach with students through LS positively affected teachers’ problem solving understandings and beliefs and their commitment to incorporating TTP practices in their future practice. Hence, this study showcases the potential role of collaborative, school-based professional development in supporting the implementation of upcoming reform proposals (Dooley et al., 2014 ; NCCA, 2016 , 2017 , 2020 ), in challenging existing beliefs and practices and fostering opportunities for teachers to work collaboratively to trial reform teaching practices over an extended period (Cajkler et al., 2015 ; Dudley et al., 2019 ). Equally, this study confirms and extends previous studies that identify time as an immense barrier to TTP. Given teachers’ positivity regarding the impact of the TTP approach, their consistent acknowledgement of the unsustainability of the unreasonable planning demands associated with TTP strengthens previous calls for the development of quality support materials in order to avoid resistance to TTP (Clarke et al., 2014 ; Takahashi, 2016 ).

The researchers are aware that while the reported changes in teachers’ problem solving beliefs and understandings are a necessary first step, for significant and lasting change to occur, classroom practice must change (Sakshaug & Wohlhuter, 2010 ). While it was intended that the MTEs would work alongside interested teachers and schools to engage further in TTP in the school term immediately following this research and initial contact had been made, plans had to be postponed due to the commencement of the COVID 19 pandemic. The MTEs are hopeful that it will be possible to pick up momentum again and move this initiative to its natural next stage. Future research will examine these teachers’ perceptions of TTP after further engagement and evaluate the effects of more regular opportunities to engage in TTP on teachers’ problem solving practices. Another possible focus is teachers’ receptiveness to TTP when quality support materials are available.

In practical terms, in order for teachers to fully embrace TTP practices, thus facilitating their students to avail of the many benefits accrued from engagement, teachers require access to professional development (such as LS) that incorporates collaboration and classroom implementation at a local level. However, quality school-based professional development alone is not enough. In reality, a TTP approach cannot be sustained unless teachers receive access to quality TTP resources alongside formal collaboration time.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education., 59 (5), 389–408.

Article   Google Scholar  

Cai, J. (2003). What research tells us about teaching mathematics through problem solving. In F. Lester (Ed.), Research and issues in teaching mathematics through problem solving (pp. 241–255). National Council for Curriculum and Assessment.

Google Scholar  

Cai, J., & Lester, F. (2010). Why is teaching through problem solving important to children learning? National Council of Teachers of Mathematics.

Cajkler, W., Wood, P., Norton, J., Pedder, D., & Xu, H. (2015). Teacher perspectives about lesson study in secondary school departments: A collaborative vehicle for professional learning and practice development. Research Papers in Education, 30 (2), 192–213.

Chapman, O. (2015). Mathematics teachers’ knowledge for teaching problem solving. LUMAT International Journal on Math Science and Technology Education, 3 (1), 19–36.

Charalambous, C. Y. (2008). Mathematical knowledge for teaching and the unfolding of tasks in mathematics lessons: Integrating two lines of research. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Proceedings of the 32nd conference of the International Group for the Psychology of Mathematics Education (pp. 281–288). PME.

Cheeseman, J. (2018). Teachers’ perceptions of obstacles to incorporating a problem solving style of mathematics into their teaching, In J. Hunter, P. Perger, & L. Darragh (Eds.), Making waves, opening spaces (Proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia) (pp. 210–217). MERGA.

Clarke, D., Cheeseman, J., Roche, A., & Van Der Schans, S. (2014). Teaching strategies for building student persistence on challenging tasks: Insights emerging from two approaches to teacher professional learning. Mathematics Teacher Education and Development, 16 (2), 46–70.

Creswell, J. W. (2009). Research design- Qualitative, quantitative and mixed methods approaches (3rd ed). Sage.

Crespo, S., & Featherstone, H. (2006). Teacher learning in mathematics teacher groups: One math problem at a time. In K. Lynch-Davis & R. L. Rider (Eds.), The work of mathematics teacher educators: Continuing the conversation (pp. 97–115). Association of Mathematics Teacher Educators.

Department of Education and Science (DES). (1999). Primary school mathematics curriculum . The Stationery Office.

Department of Education and Skills (DES). (2011). Literacy and numeracy for learning and life: The national strategy to improve literacy and numeracy among children and young people 2011–2020 . The Stationary Office.

Dooley, T., Dunphy, E., & Shiel, G. (2014). Mathematics in early childhood and primary education. Research report 18. National Council for Curriculum and Assessment.

Dudley, P., Xu, H., Vermunt, J. D., & Lang, J. (2019). Empirical evidence of the impact of lesson study on students’ achievement, teachers’ professional learning and on institutional and system evolution. European Journal of Education, 54 , 202–217. https://doi.org/10.1111/ejed.12337

Dunphy, E., Dooley, T., Shiel, G. (2014). Mathematics in early childhood and primary education. Research report 17 . National Council for Curriculum and Assessment.

Ertle, B., Chokshi, S., & Fernandez, C. (2001). Lesson planning tool. Available online: https://sarahbsd.files.wordpress.com/2014/09/lesson_planning_tool.pdf

Fernandez, C. (2005). Lesson Study: A means for elementary teachers to develop the knowledge of mathematics needed for reform-minded teaching? Mathematical Thinking and Learning, 7 (4), 265–289.

Fernandez, C., Cannon, J., & Chokshi, S. (2003). A US–Japan lesson study collaboration reveals critical lenses for examining practice. Teaching and Teacher Education, 19 , 171–185.

Flanagan, B. (2021). Teachers’ understandings of lesson study as a professional development tool, Unpublished thesis, University of Limerick.

Glaser, B. G. (1978). Theoretical sensitivity: Advances in the methodology of grounded theory . Sociology Press.

Goldenberg, E. P., Shteingold, N., & Feurzig, N. (2001). Mathematical habits of mind for young children. In F. Lester (Ed.), Research and issues in teaching mathematics through problem solving (pp. 15–30). National Council for Curriculum and Assessment.

Gutierez, S. B. (2016). Building a classroom-based professional learning community through lesson study: Insights from elementary school science teachers. Professional Development in Education, 42 (5), 801–817.

Hiebert, J. (2003). Signposts for teaching mathematics through problem solving. In F. Lester (Ed.), Research and issues in teaching mathematics through problem solving (pp. 53–62). National Council for Curriculum and Assessment.

Hourigan, M., & Leavy, A. (2015). Geometric growing patterns: What’s the rule? Australian Primary Mathematics Classroom, 20 (4), 31–40.

Hourigan, M., & Leavy, A. M. (2021). The complexities of assuming the ‘teacher of teachers’ role during Lesson Study.  Professional Development in Education. Online first.  https://doi.org/10.1080/19415257.2021.1895287

Ingram, N., Holmes, M., Linsell, C., Livy, S., McCormick, M., & Sullivan, P. (2020). Exploring an innovative approach to teaching mathematics through the use of challenging tasks: A New Zealand perspective. Mathematics Education Research Journal, 32 (3), 497–522.

Kapur, M. (2010). Productive failure in mathematical problem solving. Instructional Science, 38 , 523–550.

Klein, S., & Leikin, R. (2020). Opening mathematical problems for posing open mathematical tasks: What do teachers do and feel? Educational Studies in Mathematics, 105 , 349–365.

Koichu, B., Harel, G., & Manaster, A. (2013). Ways of thinking associated with mathematics teachers’ problem posing in the context of division of fractions. Instructional Science, 41 (4), 681–698.

Lambdin, D. V. (2003). Benefits of teaching through problem solving. In F. Lester (Ed.), Research and issues in teaching mathematics through problem solving (pp. 2–15). National Council for Curriculum and Assessment.

Leavy, A., & Hourigan, M. (2018). The role of perceptual similarity, data context and task context when selecting attributes: Examination of considerations made by 5-6 year olds in data modelling environments. Educational Studies in Mathematics. 97 (2), 163–183. https://doi.org/10.1007/s10649-017-9791-2

Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Charlotte, NC: Information Age.

Lester, F. K., Jr. (2013). Thoughts about research on mathematical problem-solving instruction. The Mathematics Enthusiast, 10 (1–2), 245–278.

Lewis, C., & Tsuchida, I. (1998). A lesson is like a swiftly flowing river: How Research lessons improve Japanese Education. American Educator, 22 (4), 12–17, 50–52.

Murata, A., Bofferding, L., Pothen, B. E., Taylor, M. W., & Wischnia, S. (2012). Making connections among student learning, content, and teaching: Teacher talk paths in elementary mathematics lesson study. Journal for Research in Mathematics Education, 43 (5), 616–650.

National Council for Curriculum and Assessment (NCCA). (2016). Background paper and brief for the development of a new primary mathematics curriculum. NCCA.

National Council for Curriculum and Assessment (NCCA). (2017). Primary mathematics curriculum. Draft specifications. Junior infant to second class. For Consultation . NCCA.

National Council for Curriculum and Assessment (NCCA). (2020). Draft primary curriculum framework. For consultation. Primary curriculum review and development. NCCA.

O’Shea, J., & Leavy, A. M. (2013). Teaching mathematical problem-solving from an emergent constructivist perspective: the experiences of Irish primary teachers. Journal of Mathematics Teacher Education, 16 (4), 293–318. https://doi.org/10.1007/s10857-013-9235-6

Polya, G. (1957). How to solve it (2nd edition) . Doubleday.

Russo, J., & Hopkins, S. (2019). Teachers’ perceptions of students when observing lessons involving challenging tasks. International Journal of Science and Mathematics Education, 17 (4), 759–779.

Russo, J., & Minas, M. (2020). Student attitudes towards learning mathematics through challenging problem solving tasks: “It’s so hard-in a good way.” International Electronic Journal of Elementary Education, 13 (2), 215–225.

Sakshaug, L. E., & Wohlhuter, K. A. (2010). Journey toward teaching mathematics through problem solving. School Science and Mathematics, 110 (8), 397–409.

Shiel, G., Kavanagh, L., & Millar, D. (2014). The national assessments of english reading and mathematics: Volume 1 performance report . Educational Research Centre.

Singer, F. M., & Voica, C. (2013). A problem-solving conceptual framework and its implications in designing problem-posing tasks. Educational Studies in Mathematics, 83 (1), 9–26.

Stacey, K. (2018). Teaching Mathematics through Problem Solving. Numeros, 98 , 7–18.

Stake, R. E. (1995). The art of case study research . Thousand Oaks, CA: Sage Publications.

Stipek, D. J., Givvin, K. B., Salmon, J. M., & MacGyvers, V. L. (2001). Teachers’ beliefs and practices related to mathematics instruction. Teaching and Teacher Education, 17 (2), 213–226.

Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed). Sage.

Sullivan, P., Askew, M., Cheeseman, J., Clarke, D., Mornane, A., & Roche, A. (2015). Supporting teachers in structuring mathematics lessons involving challenging tasks. Journal of Mathematics Teacher Education, 18 (2), 123–140.

Sullivan, P., Bobis, J., Downton, A., Feng, M., Hughes, S., Livy, S., McCormick, M., & Russo, J. (2021). An instructional model to support planning and teaching student centred structured inquiry lessons. Australian Primary Mathematics Classroom, 26 (1), 9–13.

Sullivan, P., Clarke, D., Cheeseman, J., Mornane, A., Roche, A., Sawatzki, C., & Walker, N. (2014). Students’ willingness to engage with mathematical challenges: Implications for classroom pedagogies. In J. Anderson, M. Cavanagh, & A. Prescott (Eds.), Curriculum in focus: Research guided practice. (Proceedings of the 37th annual conference of the Mathematics Education Research Group of Australasia) (pp. 597–604). MERGA.

Sullivan, P., Clarke, D. M., Clarke, B., & O’Shea, H. (2010). Exploring the relationship between task, teacher actions, and student learning. PNA, 4 (4), 133–142.

Suter, W. N. (2012). Introduction to educational research: A critical thinking approach (2nd ed). Sage.

Swan, M. (2006). Designing and using research instruments to describe the beliefs and practices of mathematics teachers. Research in Education, 75 (1), 58–70.

Takahashi, A. (2006). Characteristics of Japanese mathematics lessons. Paper presented at the APEC International Conference on Innovative Teaching Mathematics through Lesson Study, Tokyo, Japan, January 14–20. https://www.criced.tsukuba.ac.jp/math/sympo_2006/takahashi.pdf

Takahashi, A. (2008). Beyond show and tell: neriage for teaching through problem-solving—ideas from Japanese problem-solving approaches for teaching mathematics. Paper presented at the 11th International Congress on Mathematics Education in Mexico (Section TSG 19: Research and Development in Problem Solving in Mathematics Education), Monteree, Mexico.

Takahashi, A. (2016). Recent trends in Japanese mathematics textbooks for elementary grades: Supporting teachers to teach mathematics through problem solving. Universal Journal of Educational Research, 4 (2), 313–319.

Takahashi, A., Lewis, C., & Perry, R. (2013). US lesson study network to spread teaching through problem solving. International Journal for Lesson and Learning Studies, 2 (3), 237–255.

Takahashi, A., & McDougal, T. (2016). Collaborative lesson research: Maximizing the impact of lesson study. ZDM: Mathematics Education, 48, 513–526.

Thompson, A. G. (1985). Teachers’ conceptions of mathematics and the teaching of problem solving. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 281–294). Erlbaum.

Watanabe, T. (2001). Anticipating children’s thinking: A Japanese approach to instruction . National Council for Curriculum and Assessment.

Watson, A., & Ohtani, M. (2015). Task design in mathematics education: An ICMI study 22 . Springer International.

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The authors acknowledge the participating teachers’ time and contribution to this research study.

This work was supported by the Supporting Social Inclusion and Regeneration in Limerick’s Programme Innovation and Development Fund.

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Hourigan, M., Leavy, A.M. Elementary teachers’ experience of engaging with Teaching Through Problem Solving using Lesson Study. Math Ed Res J 35 , 901–927 (2023). https://doi.org/10.1007/s13394-022-00418-w

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CLAT 2025: Problem solving to critical thinking — how syllabus overhaul will affect scores

The basic five sections—english, general awareness, law, logical reasoning, and mathematics—have been retained, but the pattern of the questions has been set afresh..

— Amitendra Kumar 

The Common Law Admission Test (CLAT) has undergone significant modifications in recent years. This shift has redesigned the exam’s structure and preparation, ultimately influencing candidates’ scores. As students and educators adjust to these changes, the exam now emphasizes more on critical thinking and real-world problem-solving.

What’s the CLAT pattern?  

The major shift in the CLAT syllabus shows that the number of questions has been reduced. Earlier, the test included 200 questions which changed to 150, that had to be solved in 120 min, and this format promoted the urgent approach based on knowledge memorization. However, in the CLAT 2024 paper, the total number of questions was reduced to 120 while the time limit remained the same.

This shift also implies that instead of content coverage, students are supposed to be more interested in content comprehension, and each of them contains more comprehension levels.

The basic five sections — English, General Awareness, Law, Logical Reasoning, and Mathematics—have been retained, but the pattern of the questions has been set afresh.

For instance, English and general knowledge sections now focus majorly on passage-based questions that evaluate the candidate’s comprehension of what has been read or written and his ability to seek information.

Out of the assumptions of legal aptitude, memorization of legal maxims and legal knowledge is no longer a part of this subject, as the questions framed in this subject are depicted based on real-life exercises, which in any case are based on the basic legal principles to be applied in practice.

Impact on Student Scores

The changes that have been carried out have affected the preparation and performance of learners to varying degrees. From memorization, critical thinking and understanding have become the focus. This has resulted in a wider spread of the scores as more students do not solely rely on cramming to achieve great scores.

Adapting to the new syllabus

To succeed in this new CLAT pattern, students must adjust their preparation strategies. Rote learning is no longer enough. Instead, students should devote their time to improving their reading comprehension and critical thinking abilities. Regular practice with comprehension-based mock tests can help students become familiar with the new question styles.

Additionally, keeping up with current events, especially in legal and political fields, has become crucial for success in the general knowledge section. Reading newspapers regularly can help students stay informed and improve their holistic performance.

The CLAT syllabus overhaul has transformed the exam into a comprehensive and skills-based test. While it may pose challenges for some students, it also provides opportunities for those who excel in critical thinking and comprehension.

(The writer is a product head at Career Launcher)

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A new hybrid improved arithmetic optimization algorithm for solving global and engineering optimization problems.

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Zhang, Y.; Xing, L. A New Hybrid Improved Arithmetic Optimization Algorithm for Solving Global and Engineering Optimization Problems. Mathematics 2024 , 12 , 3221. https://doi.org/10.3390/math12203221

Zhang Y, Xing L. A New Hybrid Improved Arithmetic Optimization Algorithm for Solving Global and Engineering Optimization Problems. Mathematics . 2024; 12(20):3221. https://doi.org/10.3390/math12203221

Zhang, Yalong, and Lining Xing. 2024. "A New Hybrid Improved Arithmetic Optimization Algorithm for Solving Global and Engineering Optimization Problems" Mathematics 12, no. 20: 3221. https://doi.org/10.3390/math12203221

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• Published: 14 October 2024

A simple, effective and high-precision boundary meshfree method for solving 2D anisotropic heat conduction problems with complex boundaries

• Jing Ling 1 , 2 &
• Dongsheng Yang 1 , 2  

Scientific Reports volume  14 , Article number:  23963 ( 2024 ) Cite this article

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A simple, effective and high-precision boundary meshfree method called virtual boundary meshfree Galerkin method (VBMGM) is used to tackle 2D anisotropic heat conduction problems with complex boundaries. Temperature and heat flux are expressed by virtual boundary element method. The virtual source function is constructed through the utilization of radial basis function interpolation. Calculation model diagram and discrete model diagram of real boundaries, and schematic diagram of VBMGM are demonstrated. Using Galekin method and considering boundary conditions, the integral equation and the discrete formula of VBMGM are given in detail. The benefits of the Galerkin, meshfree, and boundary element methods are all presented in VBMGM. Seven numerical examples of general anisotropic heat conduction problems (including three numerical examples with complex boundaries and four numerical examples with mixed boundary conditions) are computed and contrasted with precise solutions and different numerical methods. The computation time of each example is given. The number of degrees of freedom used in many examples is half or less than that of the numerical method being compared. The suggested method has been demonstrated to be effective and high-precision for solving the 2D anisotropic heat conduction problems with complex boundaries.

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Introduction.

Many materials in the practical engineering are anisotropic, such as crystal 1 , wood 2 , laminar composites 3 , heat shielding materials for spacecraft 4 , and so on. Considering the heat conduction problems in anisotropic medium has many applications in scientific research and engineering, it is very important to calculate problems of anisotropic heat conductions accurately and effectively. Boundary element method (BEM) is a significant numerical technique, which has been applied by many researchers to the calculation of anisotropic heat conduction problems. BEM was implemented by Hsiao and Shiah 5 to analyze anisotropic heat conduction in aero-part turbine blades. Azis et al. 6 employed BEM for anisotropic functionally graded materials. Using the singular BEM, Li et al. 7 resolved two-dimensional anisotropic heat conduction problems. Corfdir A and Bonnet G 8 solved anisotropic conduction problems of long cylinders by using the boundary integral method. There are also a lot of references on BEM for anisotropic heat conduction problems, which are not included here. As well as having their own advantages, the above methods also have their own disadvantages, such as singular integrals and boundary layer effects, and so on. Virtual BEM 9 was given to avoid these problems. Virtual point sources were considered to distribute on virtual boundaries 9 . For better accuracy, Sun et al. 10 further developed that the virtual source functions were continuous on virtual boundaries. Xu Q et al. further obtained the virtual boundary least squares method with the continuous virtual source functions 11 , 12 .

For anisotropic heat conduction problems, the meshfree method with high precision is also an important numerical method. A dual reciprocity method and fundamental solution method were adopted to solve the anisotropic heat conduction problem 13 . Meshfree fragile points methods were employed for the complex anisotropic heat conduction problems 14 . The anisotropic stationary heat conduction problems were calculated by the meshfree radial basis function method 15 . The local semi-analytical meshfree method 16 , the meshfree method with the fundamental solutions 17 , and meshfree with the localized Chebyshev collocation method 18 were used to handle with the heat conductions. According to Yang and Ling 19 , the virtual boundary meshfree Galerkin method (VBMGM) combines the advantages of Galerkin, virtual boundary element and meshfree methods. This method has the advantages of unique weighting coefficients and symmetric coefficient matrix. Therefore, the method is extended to calculate the general anisotropic heat conduction problems.

Listed below are the remaining papers. In Section “ Governing equation and fundamental solutions for 2D anisotropic heat conduction problems with complex boundaries ”, governing equation, fundamental solutions of temperature and heat flux are presented for anisotropic heat conduction problems with complex boundaries. In Section “ Calculating formula of VBMGM for 2D anisotropic heat conduction problems with complex boundaries ”, VBMGM is specifically derived for the problem of the general anisotropic heat conduction. In Section “ Numerical examples ”, seven numerical examples are calculated, namely a square domain with variable boundary conditions, a circular domain with variable boundary conditions, a circular domain with mixed boundary conditions, an irregular-shaped domain with variable boundary conditions, an irregular-shaped domain with mixed boundary conditions, an irregular connected domain with mixed boundary conditions, a rectangular domain with mixed boundary conditions. Section 4 gives several remarks.

Governing equation and fundamental solutions for 2D anisotropic heat conduction problems with complex boundaries

Governing equation is given the anisotropic heat conduction problems 15 , namely

here, \(\varOmega\) is the computational domain; \({\varvec{x}}\) is the computational point; \(T({\varvec{x}})\) is temperature. i , j  = 1, 2. \({x_i}\) is the coordinate axis in the i th direction. \({k_{ij}}\) is the heat conduction coefficients of the general anisotropic heat conduction problems.

Temperature fundamental solution \({T^*}\) 20 , 21 of the 2D anisotropic heat conduction problems is

here, \(\left| {{k_{ij}}} \right|={k_{11}}{k_{22}} - k_{{12}}^{2}\) ; \(\varvec{\zeta}\) is the source point; \(r({\varvec{x}},\varvec{\xi})=({k_{22}}{r_1}^{2} - {k_{12}}{r_1}{r_2} - {k_{21}}{r_1}{r_2}\) \(+{k_{11}}r_{2}^{2}{)^{0.5}}/{\left| {{k_{ij}}} \right|^{0.5}}\) . \({r_1}={x_1} - {\zeta _1}\) and \({r_2}={x_2} - {\zeta _2}\) .

Heat flux fundamental solution \({q^*}\) is

here, \({n_i}\) is the normal cosine of \({x_i}\) direction.

Calculating formula of VBMGM for 2D anisotropic heat conduction problems with complex boundaries

The calculation model for anisotropic heat conduction problems with complex boundaries is displayed in Fig.  1 . The boundary conditions of the anisotropic heat conduction problems are

where \(\varGamma _{T}\) and \(\varGamma _{q}\) are the predetermined boundaries of temperature and heat flux; \(\bar {T}({\varvec{x}})\) is the predetermined temperature; \(\bar {q}({\varvec{x}})\) is the predetermined heat flux.

Calculation model of 2D anisotropic heat conduction problems with complex boundaries.

Discrete model of real boundaries.

Considering the Galerkin method, and Eqs. ( 4 )–( 5 ) boundary conditions, the VBMGM integral equation for anisotropic heat conduction problems with complex boundaries 19 should be obtained

where the weighted coefficients \({w_T}=\delta T({\varvec{x}})\) , and \({w_q}=\delta q({\varvec{x}})\) .

\(\varGamma _{T}\) and \(\varGamma _{q}\) in Fig.  2 can be discretized. Equation ( 6 ) is converted into the following expression

here \({e_T}\) and \({e_q}\) denote the number of elements into which \({\varGamma _T}\) and \({\varGamma _q}\) are divided; \({e_T}\) and \({e_q}\) are Gaussian points in real elements of \({\varGamma _T}\) and \({\varGamma _q}\) ; \({\varvec{x}}_{i}^{j}\) is the \(j{\text{th}}\) Gaussian point within the \(i{\text{th}}\) element; \(w({\varvec{x}}_{i}^{j})\) is the Gaussian integration coefficient; \({J_T}\) and \({J_T}\) are the element Jacobians, respectively.

Schematic diagram of virtual BEM.

Virtual BEM can be applied to depict \(T({\varvec{x}})\) and \(q({\varvec{x}})\) . The schematic diagram of the virtual BEM is presented for anisotropic heat conduction problems with complex boundaries in Fig.  3 . Virtual boundary S of virtual domain \(\varOmega ^\prime\) can be obtained to extend outward through the outer normal of the points on real boundaries. It is assumed that the continuous virtual source function \(\beta (\varvec{\zeta})\) can be found on the virtual node \(\varvec{\zeta}\) of virtual boundary S . S is discretized into \({e_v}\) virtual elements. Virtual node number is Nv on S . Considering virtual source function \(\beta (\varvec{\zeta})\) is approximated using radial basis function interpolation, \(T({\varvec{x}})\) and \(q({\varvec{x}})\) of Eqs. ( 4 )–( 5 ) are written

where \(\varvec{\zeta}_{{i^{\prime}}}^{{j^{\prime}}}\) refers to the \(j^{\prime}{\text{th}}\) Gaussian point within the \(i^{\prime}{\text{th}}\) virtual element; \(w(\varvec{\zeta}_{{i^{\prime}}}^{{j^{\prime}}})\) refers to the Gaussian integration coefficient of virtual element; \({J_v}\) is the virtual element Jacobian; \(N(\varvec{\zeta}_{{i^{\prime}}}^{{j^{\prime}}},{\varvec{\zeta}_s})\) (Eq. ( 11 ) 19 ) is the shape function of \({\varvec{\zeta}_s}.\)   m is the virtual node number within compactly supported domain of \(\varvec{\zeta}_{{i^{\prime}}}^{{j^{\prime}}}\) . \(\varvec{\beta}\) , \({\varvec{A}}\) and \({\varvec{B}}\) are

here \({A_s}\) and \({B_s}\) of Eqs. ( 11 )–( 12 ) are

here, Fig.  3 illustrates a compactly supported domain of \({\varvec{\zeta}_k}\) with K Gaussian points.

Considering Eqs. ( 8 )–( 9 ), ( 7 ) can changed into

Note \({{\partial T({\varvec{x}})} \mathord{\left/ {\vphantom {{\partial T({\varvec{x}})} {\partial {\beta _s}}}} \right. \kern-0pt} {\partial {\beta _s}}}={A_s}\) and \({{\partial q({\varvec{x}})} \mathord{\left/ {\vphantom {{\partial q({\varvec{x}})} {\partial {\beta _s}}}} \right. \kern-0pt} {\partial {\beta _s}}}={B_s}\) of Eqs. ( 8 )–( 9 ). Then, Eq. ( 15 ) can be expressed as a matrix

here the known symmetric coefficient matrix \({\varvec{C}}{\text{=}}{[{C_{st}}]_{Nv \times Nv}}\) and the known right-side matrix \({\varvec{D}}={[{D_s}]_{Nv}}\) are

\({w_T}=\delta T({\varvec{x}})={A_s}\) , \({w_q}=\delta q({\varvec{x}})={B_s}\) in Eq. ( 15 ).

In order to better illustrate the implementation process of the method in this paper, the flowchart is given in Fig.  4 .

The flowchart for the implementation.

Numerical examples

Example 1. a square domain with variable boundary conditions.

A general anisotropic square domain (1 m × 2 m) with boundary conditions is demonstrated in Fig.  5 . Boundary conditions are \({\bar {T}_1}= - x_{2}^{2}+2^\circ C\) , \({\bar {T}_2}=x_{1}^{2}/2+{x_1}+1^\circ C\) , \({\bar {T}_3}= - x_{2}^{2}+{x_2}+2.5^\circ C\) and \({\bar {T}_4}=x_{1}^{2}/2+2^\circ C\) on real boundaries \({\varGamma _1}\) , \({\varGamma _2}\) , \({\varGamma _3}\) and \({\varGamma _4}\) , respectively. General anisotropic heat conduction coefficients are \({k_{11}}=1\text{W}/(\text{m} \cdot ^\circ \text{C})\) , \({k_{12}}={k_{21}}=0.5\text{W}/(\text{m} \cdot ^\circ \text{C}),\) and \({k_{22}}=1\text{W}/(\text{m} \cdot ^\circ \text{C}).\)

The numerical discrete diagram of this computational example is shown in Fig.  6 . Each real boundary \({\varGamma _1},\) \({\varGamma _2}\) , \({\varGamma _3}\) or \({\varGamma _4}\) has 20 real elements. The circular virtual boundary S is divided into 16 elements. The center coordinates of S are (0.5, 0.5). The radius of S is \(r^{\prime}=2\text{m}\) . At the same time, 16 virtual nodes are placed on the virtual boundary S . The virtual node is located at the center of the virtual element. 10 virtual nodes are used to interpolate the virtual source function. 4 Gaussian points are used to integrate each virtual or real element.

The mechanical quadrature methods (MQMs) 22 were used to solve this problem with 32 computational degrees of freedom. It should have been noted that this method uses 16 computational degrees of freedom. The analytical solution of this numerical example 22 is \(T=x_{1}^{2}/2+{x_1}{x_2} - x_{2}^{2}+2\) . The absolute errors of MQMs 22 are 7.925 × 10 − 4 , 3.209 × 10 − 3 and 3.885 × 10 − 4 at points (0.1,0.1), (0.2,0.2) and (0.6,0.6). The calculation results and corresponding absolute errors are displayed in Table  1 . The calculation time of this method is 0.468 s. The paper presents the accuracy of the method in this paper.

In order to prove the stability of the proposed method for the general anisotropic heat conduction problems, two cases are given. Case 1 : the distance ( \(d=r^{\prime}/(0.5\sqrt 2 ) - 1\) ) between the virtual and real boundaries is changed. The computational results are given in Table  2 . Case 2 : different Gaussian points on the virtual and real elements are selected. The computational results are listed in Table  3 .

Calculation model of Example 1.

Discrete diagram for Example 1.

Example 2. A circular domain with variable boundary conditions

A general anisotropic circular domain (the radius R  = 1 m) with variable boundary conditions is demonstrated in Fig.  7 . r is the distance from any point in the domain to the center of the circle. θ is the angle between r and the positive half axis of x 1 . Boundary condition is \(\bar {T}({x_1},{x_2})=x_{1}^{3}/5 - x_{1}^{2}{x_2}+{x_1}x_{2}^{2}+x_{2}^{3}{\text{/3 }}^\circ C\) ( \(0^\circ \leqslant \theta \leqslant 360^\circ\) ) on real boundary \(\varGamma\) . General anisotropic heat conduction coefficients are \({k_{11}}=5\text{W}/(\text{m} \cdot ^\circ \text{C})\) , \({k_{12}}={k_{21}}=2\text{W}/(\text{m} \cdot ^\circ \text{C})\) , and \({k_{22}}=1\text{W}/(\text{m} \cdot ^\circ \text{C})\) .

Calculation model of Example 2.

Discrete diagram for Example 2.

The numerical discrete diagram of this computational example is shown in Fig.  8 . The circular real boundary \(\varGamma\) has 40 real elements. The circular virtual boundary S is divided into 40 elements. The center coordinates of S are (0, 0). The radius of S is \(r^{\prime}=1.2\text{m}\) . At the same time, 40 virtual nodes are placed on the virtual boundary S . The virtual node is located at the center of the virtual element. 5 virtual nodes are used to interpolate the virtual source function. 4 Gaussian points are used to integrate each virtual or real element.

Constant boundary element method (CBEM), linear boundary element method (LBEM), discontinuous linear boundary element method (DLBEM), quadratic boundary element method (QBEM), discontinuous quadratic boundary element method (DQBEM) were used to solve this numerical problem with 40 elements 23 . The exact solution of temperature is \(T({x_1},{x_2})=x_{1}^{3}/5 - x_{1}^{2}{x_2}+{x_1}x_{2}^{2}+x_{2}^{3}{\text{/3 }}^\circ C\) . The degrees of freedom of CBEM, LBEM, DLBEM, QBEM, and DQBEM were 40, 80, 80, 120, and 120, respectively. Note that the number of degrees of freedom used in this paper is the number of virtual nodes, namely 40. The temperature at the point (0.25 m, 0.25 m) is calculated and listed in Table  4 . Temperatures with different radius ( r  = 0.8 m, 0.6 m, 0.4 m, and 0.2 m) are obtained in Table  5 . The calculation time of this method is 0.188 s. The accuracy of the results of this numerical example is validated.

Example 3. A circular domain with mixed boundary conditions

In order to verify that the method in this paper is able to compute anisotropic heat conduction problems with mixed boundary conditions, boundary conditions of Example 2 are altered. The anisotropic heat conduction coefficients are the same as Example 2. Boundary conditions are \(\bar {q}({x_1},{x_2})= - (x_{1}^{2} - 6x_{1}{x_2}+7x_{2}^{2}){n_1} - (x_{1}^{2}/5 - 2x_{1}{x_2}+3x_{2}^{2}){n_2}\) ( \(0^\circ \leqslant \theta \leqslant 180^\circ\) ) and \(\bar {T}({x_1},{x_2})=x_{1}^{3}/5 - x_{1}^{2}{x_2}+{x_1}x_{2}^{2}+x_{2}^{3}{\text{/3 }}^\circ C\) ( \(180^\circ \leqslant \theta \leqslant 360^\circ\) ). Here the number of degrees of freedom is selected to be the same as the number of degrees of freedom for DLBEM 23 , namely 80 virtual nodes on virtual boundary S . The number of elements on both virtual and real boundaries is 80. The remaining conditions of discretization are the same as in Example 2. The temperature value and percentage error are 0.008317 and 0.19% at the point (0.25 m, 0.25 m) by the method of this paper. Temperatures with different radius ( r  = 0.8 m, 0.6 m, 0.4 m, and 0.2 m) are also given in Table  6 . The calculation time of this method is 0.550 s. Similarly, the accuracy of the results of this numerical example is proved.

Example 4. An irregular-shaped domain with variable boundary conditions

The real boundary of the irregular-shaped domain is demonstrated in Fig.  9 . r is the distance from any point on the real boundary \(\varGamma\) to the coordinate origin. The coordinates of any point on the boundary of the irregular shape are \((r\cos \theta ,r\sin \theta )\) . \(r=[1+2n+{n^2} - (n+1)\cos (4\theta )]/{n^2}\) . n is the number of corners of irregular shape. In this example n  = 4. θ ( \(0^\circ \leqslant \theta \leqslant 360^\circ\) ) is the angle between r and the positive half axis of x 1 . Boundary condition is

General anisotropic heat conduction coefficients are \({k_{11}}=3\text{W}/(\text{m} \cdot ^\circ \text{C})\) , \({k_{12}}={k_{21}}=1.5\text{W}/(\text{m} \cdot ^\circ \text{C})\) , and \({k_{22}}=2\text{W}/(\text{m} \cdot ^\circ \text{C})\) .

Calculation model of Example 4.

Discrete diagram for Example 4.

The numerical discrete diagram of this computational example is shown in Fig.  10 . The real boundary \(\varGamma\) has 180 real elements, namely take one real element every two degrees. Each straight-line virtual boundary \({S_1}\) , \({S_2}\) , \({S_3}\) or \({S_4}\) is divided into 20 elements. The four corner coordinates of the virtual boundary are A (-2, 2), B (2, 2), C (2, -2), D (-2, -2). At the same time, 20 virtual nodes are placed on the each straight-line virtual boundary \({S_1}\) , \({S_2}\) , \({S_3}\) or \({S_4}\) . The virtual node is located at the center of the virtual element. 5 virtual nodes are used to interpolate the virtual source function. 2 Gaussian points are used to integrate each virtual or real element.

Temperatures with different radius ( r  = 1.2 m, 0.8 m, and 0.4 m) are obtained in Table  4 . The exact solution of this numerical example 24 is also.

The meshfree radial basis integral equation method 24 was employed to solve this numerical example with 145 nodes. The root mean square error was between 10 − 1 and 10 − 2 24 . Note that the number of degrees of freedom is the number of virtual boundary nodes in this method, namely 80. The root mean square error is 2.60 × 10 − 7 according to Table  7 . The calculation time of this method is 0.293 s. The accuracy of the results of this numerical example is proved.

Example 5. An irregular-shaped domain with mixed boundary conditions

Similarity, in order to verify that the method in this paper is able to compute irregular-shaped anisotropic heat conduction problems of with mixed boundary conditions, boundary conditions of Example 4 are changed. The temperature boundary condition \(\bar {T}({x_1},{x_2})\) is imposed on real boundary \(\partial \Omega\) ( \(0^\circ \leqslant \theta \leqslant 180^\circ\) ). The heat flux boundary condition \(\bar {q}({x_1},{x_2})\) is applied to real boundary \(\partial \Omega\) ( \(180^\circ \leqslant \theta \leqslant 360^\circ\) ). \(\bar {T}({x_1},{x_2})\) and \(\bar {q}({x_1},{x_2})\) are

here, \({n_i}\) is the normal cosine of \({x_i}\) direction. \({k_{11}}=3\) , \({k_{12}}={k_{21}}=1.5\) , and \({k_{22}}=2\) . \(\partial \bar {T}/\partial {x_1}\) and \(\partial \bar {T}/\partial {x_2}\) are

The numerical discretization process is the same as Example 4. Similarity, temperatures with different radius ( r  = 1.2 m, 0.8 m, and 0.4 m) are given in Table  8 . The root mean square error is 2.43 × 10 − 7 according to Table  8 . The calculation time of this method is 0.276 s. The accuracy of the results for the irregular-shaped domain with mixed boundary conditions is validated.

Example 6. An irregular connected domain with mixed boundary conditions

An irregular connected domain with mixed boundary conditions is demonstrated in Fig.  11 . r is the distance from any point on the real boundary to the coordinate origin. The coordinates of any point on the boundary of the irregular shape are \((r\cos \theta ,r\sin \theta )\) . \(r={e^{\sin \theta }}{\sin ^2}(2\theta )+{e^{\cos \theta }}{\cos ^2}(2\theta )\) . θ ( \(0^\circ \leqslant \theta \leqslant 360^\circ\) ) is the angle between r and the positive half axis of x 1 . General anisotropic heat conduction coefficients are \({k_{11}}=2\text{W}/(\text{m} \cdot ^\circ \text{C})\) , and \({k_{12}}={k_{21}}={k_{22}}=1\text{W}/(\text{m} \cdot ^\circ \text{C})\) . Temperature boundary conditions on \({\varGamma _1}\) are \(\bar {T}({x_1},{x_2})={e^{\left( {{x_1} - {x_2}} \right)}}\cos \left( {{x_2}} \right)\) . Heat flux boundary conditions on \({\varGamma _2}\) are

here, \({n_i}\) is the normal cosine of \({x_i}\) direction. \(\partial \bar {T}/\partial {x_1}\) and \(\partial \bar {T}/\partial {x_2}\) are

Calculation model of Example 6.

Discrete diagram for Example 6.

The center coordinates of \({\varGamma _2}\) is (0.5, 0.5). The radius of \({\varGamma _2}\) is 0.4.

The numerical discrete diagram of this computational example is shown in Fig.  12 . The real boundary \({\varGamma _1}\) has 180 real elements, namely take one real element every two degrees. There are 60 real elements on the real boundary \({\varGamma _2}\) . Both the outer and inner circular virtual boundaries S 1 and S 2 are divided into 50 virtual elements. The center coordinates of S 1 and S 2 are (0.5, 0.5). The radii of S 1 and S 2 are 5 and 0.3, respectively. The virtual node is located at the center of the virtual element. 5 virtual nodes are used to interpolate the virtual source function. 2 Gaussian points are used to integrate each virtual or real element.

Temperatures and heat fluxes ( r  = 0.55 m, the angle interval 30 degrees) are given in Table  9 . The root mean square error of temperatures is 6.32 × 10 − 5 . This numerical example is also computed by singular boundary method (SBM) and boundary element method (BEM) 20 . Note that the number of degrees of freedom is the number of virtual boundary nodes, namely 100. The calculation time of this method is 0.687 s. In the same calculation of degrees of freedom, the root mean square errors of temperatures of SBM and BEM are is between 10 − 2 and 10 − 4 . The accuracy of the results of this numerical example is proved.

Example 7. A rectangular domain with mixed boundary conditions

A rectangular domain (1 m × 2 m) with boundary conditions is shown in Fig.  13 . Boundary conditions are \({\bar {T}_1}=400^\circ C\) , \({\bar {q}_2}=0\) , \({\bar {T}_3}=100^\circ C\) and \({\bar {q}_4}=0\) on real boundaries \({\varGamma _1}\) , \({\varGamma _2}\) , \({\varGamma _3}\) and \({\varGamma _4}\) , respectively. General anisotropic heat conduction coefficients are \({k_{11}}=5\text{W}/(\text{m} \cdot ^\circ \text{C})\) , \({k_{12}}={k_{21}}=1\text{W}/(\text{m} \cdot ^\circ \text{C})\) and \({k_{22}}=2\text{W}/(\text{m} \cdot ^\circ \text{C})\) .

Calculation model of Example 7.

Discrete diagram for Example 7.

The numerical discrete diagram of this computational example is shown in Fig.  14 . Real boundaries \({\varGamma _1}\) , \({\varGamma _2}\) , \({\varGamma _3}\) or \({\varGamma _4}\) have 40, 20, 40, and 20 real elements, respectively. There are 30, 15, 30, and 15 virtual elements, respectively. The virtual node is located at the center of the virtual element. 8 virtual nodes are used to interpolate the virtual source function. 4 Gaussian points are used to integrate each virtual or real element.

Finite element method (FEM) is used to solved this problem with 200 elements by COMSOL software in Fig.  15 . COMSOL software shows that the number of degrees of freedom used is 861. Note that 90 computational degrees of freedom are used by this method, namely the number of virtual nodes. Calculation results of x 2  = 0.5 m, 0, and − 0.5 m are given in Table  10 . The calculation results of FEM are used as a benchmark, and the corresponding relative errors are also calculated in Table  10 . The calculation time of this method is 0.986 s. The accuracy of the results of this numerical example is proved.

Discrete graph of finite element method by COMSOL software.

Conclusions

The virtual boundary meshfree Galerkin method (VBMGM) as a useful boundary-type meshfree method is given for 2D anisotropic heat conduction problems with complex boundaries. Seven numerical examples with variable boundary conditions, mixed boundary conditions, or the irregular-shaped domain are calculated. The effectiveness and high accuracy for anisotropic heat conduction problems with complex boundaries are testified. Temperature and heat flux are expressed by virtual boundary element method. The virtual source functions on virtual boundaries are approximated by radial basis function interpolation. The integral equation and the discrete formula are derived for anisotropic heat conduction problems with complex boundaries by using the Galerkin method. Simultaneously, calculation model diagram and discrete model diagram of real boundaries, and schematic diagram are demonstrated in the process of formula derivation. VBMGM thus combines the benefits of Galerkin, meshfree, and boundary element techniques.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Liu, Q., Yang, H.-N., Li, T., Tian, H. & Yang, Z. An efficient and unified method for band structure calculations of 2D anisotropic photonic-crystal fibers. Calcolo 61 , 20. https://doi.org/10.1007/s10092-024-00572-6 (2024).

Article   MathSciNet   Google Scholar  

Shi, X., Bi, R., Wan, Z., Jiang, F. & Rojas, O. J. Solid wood modification toward anisotropic elastic and insulative foam-like materials. ACS Nano 18 , 7959–7971. https://doi.org/10.1021/acsnano.3c10650 (2024).

Article   PubMed   Google Scholar  

Mitrou, A., Arteiro, A., Reinoso, J. & Camanho, P. P. Modeling fracture of multidirectional thin-ply laminates using an anisotropic phase field formulation at the macro-scale. Int. J. Solids Struct. 273 , 112221. https://doi.org/10.1016/j.ijsolstr.2023.112221 (2023).

Article   Google Scholar  

Leonov VV, Zarubin VS, Ayrapetyan MA. Application efficiency analysis of anisotropic heat-shielding materials when executing trajectories with multiple entrance in the atmosphere, Moscow, Russia: 2021, p. 120010. https://doi.org/10.1063/5.0035722 .

Hsiao, Y.-F. & Shiah, Y.-C. Efficient BEM modeling of the heat transfer in the turbine blades of aero-parts. Aerospace 10 , 885. https://doi.org/10.3390/aerospace10100885 (2023).

Azis, M. I., Toaha, S., Hamzah, S. & Solekhudin, I. A numerical investigation of 2D transient heat conduction problems in anisotropic FGMs with time-dependent conductivity. J. Comput. Sci. 73 , 102122. https://doi.org/10.1016/j.jocs.2023.102122 (2023).

Li, W., Xu, S. & Shao, M. Simulation of two-dimensional steady-state heat conduction problems by a fast singular boundary method. Eng. Anal. Bound. Elem. 108 , 149–157. https://doi.org/10.1016/j.enganabound.2019.06.020 (2019).

Corfdir, A. & Bonnet, G. Strategies using boundary integral equations for solving exterior anisotropic conduction problems outside long cylinders. Eng. Anal. Bound. Elem. 76 , 1–9. https://doi.org/10.1016/j.enganabound.2016.12.005 (2017).

Burgess, G. & Mahajerin, E. A comparison of the boundary element and superposition methods. Comput. Struct. 19 , 697–705. https://doi.org/10.1016/0045-7949(84)90169-X (1984).

Sun, H., Zhang, L., Xu, Q. & Zhang, Y. Nonsingularity Boundary Element Methods (Dalian University of Technology Press, 1999).

Google Scholar  

Xu, Q., Zhang, Z. & Si, W. Virtual boundary meshless least square collocation method for calculation of 2D multi-domain elastic problems. Eng. Anal. Bound. Elem. 36 , 696–708. https://doi.org/10.1016/j.enganabound.2011.12.008 (2012).

Yang, D.-S. & Xu, Q. Virtual boundary meshless least square integral method with moving least squares approximation for 2D elastic problem. Eng. Anal. Bound. Elem. 37 , 616–623. https://doi.org/10.1016/j.enganabound.2013.01.006 (2013).

Protektor, D. O., Kolodyazhny, V. M., Lisin, D. O. & Lisina, O. Y. A meshless method of solving three-dimensional nonstationary heat conduction problems in anisotropic materials. Cybern. Syst. Anal. 57 , 470–480. https://doi.org/10.1007/s10559-021-00372-8 (2021).

Guan, Y. & Atluri, S. N. Meshless fragile points methods based on Petrov-Galerkin weak-forms for transient heat conduction problems in complex anisotropic nonhomogeneous media. Numerical Meth Engineering 122 , 4055–4092. https://doi.org/10.1002/nme.6692 (2021).

Reutskiy, S. Y. A meshless radial basis function method for 2D steady-state heat conduction problems in anisotropic and inhomogeneous media. Eng. Anal. Bound. Elem. 66 , 1–11. https://doi.org/10.1016/j.enganabound.2016.01.013 (2016).

Wang, C., Wang, F. & Gong, Y. Analysis of 2D heat conduction in nonlinear functionally graded materials using a local semi-analytical meshless method. MATH 6 , 12599–618. https://doi.org/10.3934/math.2021726 (2021).

Wang, F. A localized space-time method of fundamental solutions for diffusion and convection-diffusion problems. AAMM 12 , 940–958. https://doi.org/10.4208/aamm.OA-2019-0269 (2020).

Wang, F., Zhao, Q., Chen, Z. & Fan, C.-M. Localized Chebyshev collocation method for solving elliptic partial differential equations in arbitrary 2D domains. Appl. Math. Comput. 397 , 125903. https://doi.org/10.1016/j.amc.2020.125903 (2021).

Yang, D.-S. & Ling, J. Calculating the single-domain heat conduction with heat source problem by virtual boundary meshfree Galerkin method. Int. J. Heat Mass Transf. 92 , 610–616. https://doi.org/10.1016/j.ijheatmasstransfer.2015.09.015 (2016).

Article   ADS   Google Scholar  

Gu, Y., Chen, W. & He, X.-Q. Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media. Int. J. Heat Mass Transf. 55 , 4837–4848. https://doi.org/10.1016/j.ijheatmasstransfer.2012.04.054 (2012).

Marin, L. An invariant method of fundamental solutions for two-dimensional steady-state anisotropic heat conduction problems. Int. J. Heat Mass Transf. 94 , 449–464. https://doi.org/10.1016/j.ijheatmasstransfer.2015.11.080 (2016).

Luo, X. & Huang, J. High-accuracy quadrature methods for solving boundary integral equations of steady-state anisotropic heat conduction problems with Dirichlet conditions. Int. J. Comput. Math. 91 , 1097–1121. https://doi.org/10.1080/00207160.2013.828050 (2014).

Mera, N. S., Elliott, L., Ingham, D. B. & Lesnic, D. A comparison of boundary element method formulations for steady state anisotropic heat conduction problems. Eng. Anal. Bound. Elem. 25 , 115–128. https://doi.org/10.1016/S0955-7997(00)00050-3 (2001).

Ooi, E. H., Ooi, E. T. & Ang, W. T. Numerical investigation of the meshless radial basis integral equation method for solving 2D anisotropic potential problems. Eng. Anal. Bound. Elem. 53 , 27–39. https://doi.org/10.1016/j.enganabound.2014.12.004 (2015).

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This work was supported by the National Natural Science Foundation of China (No. 11762005), Science and Technology Program of Guizhou Province (Qianke He Foundation-ZK [2021] Key 021), and Rolling Supported Provincial University Scientific Research Platform Project of Guizhou Provincial Department of Education (Qian Jiaoji [2022] Key 012).

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Jing Ling & Dongsheng Yang

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J.L.: Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing - original draft, Writing - review & editing. D.Y.: Conceptualization, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing - original draft, Writing - review & editing.

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Ling, J., Yang, D. A simple, effective and high-precision boundary meshfree method for solving 2D anisotropic heat conduction problems with complex boundaries. Sci Rep 14 , 23963 (2024). https://doi.org/10.1038/s41598-024-74950-z

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DOI : https://doi.org/10.1038/s41598-024-74950-z

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