teaching approach in problem solving

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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..

Center for Teaching Innovation

Resource library.

Establishing Community Agreements and Classroom Norms
Sample group work rubric
Problem-Based Learning Clearinghouse of Activities, University of Delaware

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. 

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 @helyn_kim.

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.

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

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

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

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

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

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

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

Where can I learn more?

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

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Problem Solving 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.

For more information...

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.

Designing Your Course
A Teaching Timeline: From Pre-Term Planning to the Final Exam
The First Day of Class
Group Agreements
Classroom Debate
Flipped Classrooms
Leading Discussions
Polling & Clickers
Teaching with Cases
Engaged Scholarship
Devices in the Classroom
Beyond the Classroom
On Professionalism
Getting Feedback
Equitable & Inclusive Teaching
Advising and Mentoring
Teaching and Your Career
Teaching Remotely
Tools and Platforms
The Science of Learning
Bok Publications
Other Resources Around Campus

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

5 Teaching Mathematics Through Problem Solving

Janet Stramel

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

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

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

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

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

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

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

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

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

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

Key features of a good mathematics problem includes:

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

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

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

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

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

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

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

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

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

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

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

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

Number Talks

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

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

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

Inside Mathematics Number Talks
Number Talks Build Numerical Reasoning

Saying “This is Easy”

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

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

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

Instead, you and your students could say the following:

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

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

Using “Worksheets”

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

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

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

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

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

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

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

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

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

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

when we teach students how to problem solve

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

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

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

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Problem-Based Learning (PBL)

What is Problem-Based Learning (PBL)? PBL is a student-centered approach to learning that involves groups of students working to solve a real-world problem, quite different from the direct teaching method of a teacher presenting facts and concepts about a specific subject to a classroom of students. Through PBL, students not only strengthen their teamwork, communication, and research skills, but they also sharpen their critical thinking and problem-solving abilities essential for life-long learning.

By working with PBL, students will:

Become engaged with open-ended situations that assimilate the world of work
Participate in groups to pinpoint what is known/ not known and the methods of finding information to help solve the given problem.
Investigate a problem; through critical thinking and problem solving, brainstorm a list of unique solutions.
Analyze the situation to see if the real problem is framed or if there are other problems that need to be solved.

How to Begin PBL

Establish the learning outcomes (i.e., what is it that you want your students to really learn and to be able to do after completing the learning project).
Find a real-world problem that is relevant to the students; often the problems are ones that students may encounter in their own life or future career.
Discuss pertinent rules for working in groups to maximize learning success.
Practice group processes: listening, involving others, assessing their work/peers.
Explore different roles for students to accomplish the work that needs to be done and/or to see the problem from various perspectives depending on the problem (e.g., for a problem about pollution, different roles may be a mayor, business owner, parent, child, neighboring city government officials, etc.).
Determine how the project will be evaluated and assessed. Most likely, both self-assessment and peer-assessment will factor into the assignment grade.

Designing Classroom Instruction

How to Orchestrate a PBL Activity

Explain Problem-Based Learning to students: its rationale, daily instruction, class expectations, grading.
Serve as a model and resource to the PBL process; work in-tandem through the first problem
Help students secure various resources when needed.
Supply ample class time for collaborative group work.
Give feedback to each group after they share via the established format; critique the solution in quality and thoroughness. Reinforce to the students that the prior thinking and reasoning process in addition to the solution are important as well.

Teacher’s Role in PBL

The Role of the Students

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

Original Article
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.

Lesson Study and Its Role in the Implementation of Curriculum Reform in China

Learning to Teach Mathematics Through Problem Solving

Avoid common mistakes on your manuscript.

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.

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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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Teaching through Problem Solving (TTP)
Problem solving
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Problem-Solving Method in Teaching

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

Definition of problem-solving method.

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

Meaning of Problem-Solving Method

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

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

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

Benefits of Problem-Solving Method

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

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

Steps in Problem-Solving Method

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

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

Verification of the concluded solution or Hypothesis

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

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

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Data Descriptor
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Published: 25 April 2024

Students’ performance, attitude, and classroom observation data to assess the effect of problem-based learning approach supplemented by YouTube videos in Ugandan classroom

Nicholus Gumisirizah 1 ,
Joseph Nzabahimana 1 &
Charles M. Muwonge 2

Scientific Data volume 11 , Article number: 428 ( 2024 ) Cite this article

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Applied physics

In response to global demands, Uganda’s Vision 2040 seeks to transform the country into a modern and prosperous nation by implementing Sustainable Development Goal (SDG) 4, focusing on equitable and quality education. The 21st-century workforce requires individuals who can effectively navigate complex workplace challenges. This dataset was gathered from Form-2 Ugandan secondary school students (aged 12 to 15) across 12 schools in the Sheema District. The dataset comprises three types of data: students’ performance in a physics topic (simple machines), their attitudes toward problem-solving and critical thinking when learning physics using Problem-Based Learning (PBL) supplemented by YouTube videos, and classroom observations documented with the reformed teaching observational protocol (RTOP). The intervention of teaching using PBL was executed in 2022, collecting data from 973 lower secondary school students. The intervention involved three approaches: one group (144 students) received PBL along with YouTube videos, another group of 482 students received PBL alone, and a third group (347 students) was taught using the traditional method. This data article explains the study’s data creation, collection, and analysis process. The dataset holds significance for secondary school teachers, policymakers, and researchers, offering insights into the impact of PBL with and without ICT resources on learning physics and students’ attitudes toward these learner-centered approaches.

A meta-analysis to gauge the impact of pedagogies employed in mixed-ability high school biology classrooms

Secondary school students’ attitude towards mathematics word problems

Dataset and validation of the approaches to study skills inventory for students

Background & summary.

Physics education in secondary schools plays a vital role in developing students’ social, physical, leadership, and problem-solving skills. Understanding physics concepts equips learners to know how things work, enabling them to apply this understanding to real-life situations 1 . The physics teaching is structured around activity-based 2 chapters and topics, emphasizing hands-on experiences 3 and practical applicability in everyday life. However, many students find physics challenging, necessitating an active teaching approach. Teaching in physics remains dynamic and interactive, with teachers adopting various strategies to engage students actively. Reciprocal teaching involves dialogues between the teacher and small student groups, while peer collaboration fosters cooperative work on class activities. Problem-Based Learning (PBL) 4 , 5 , 6 is a student-centered approach that encourages group-based learning and teacher facilitation. It has been widely adopted in various educational fields, promoting problem-solving in learning environments. Implementing PBL follows a five-stage process:

Finding a problem

The teacher prepares a task for students to investigate, stimulating problem-solving abilities.

Organizing ideas on the problem

Learners investigate the problem, generate ideas, and receive probing questions from the facilitator to stimulate critical thinking.

The teacher facilitates the distribution of learners into groups, each focusing on solving a particular problem related to the main task. Responsibilities are assigned within each group, promoting cooperation.

Present findings

Learners present solutions to the problem and receive feedback from peers, consolidating their learning outcomes.

Generalizing

Problem-solving leads to the development of skills essential for solving complex, real-world situations. These skills, including problem-solving, creativity, communication, cooperation, and innovation, prepare students to adapt to change and overcome 21st-century challenges.

Integrating YouTube videos as Information and Communication Technology (ICT) tools within a PBL approach offers a multifaceted strategy to enhance physics education 7 , 8 . High-quality videos aligned with curriculum objectives introduce real-world problems and cater to diverse learning styles. Interactive features and accessibility allow continuous learning, and educators can curate playlists to align with curriculum goals. The flipped classroom model 9 combines videos with problem-solving discussions 10 , creating a dynamic learning environment that deepens students’ understanding of physics concepts and their practical applications.

Physics is a subject that holds a significant position in promoting scientific literacy, critical thinking, and essential life skills. However, conventional teaching methods often struggle to engage and empower students in the subject matter effectively. This inadequacy is a pressing concern, as it can hinder students from developing a strong foundation in physics, which is essential for their academic and practical pursuits. This study was critically important due to the existing challenges within physics education in Ugandan secondary schools. Incorporating innovative teaching approaches, such as PBL supplemented by YouTube videos, becomes pivotal in addressing these challenges. These methods can enhance students’ comprehension of physics and nurture vital skills like problem-solving, creativity, communication, cooperation, and innovation. These skills are indispensable for students to thrive in a rapidly evolving, knowledge-driven world.

Sharing the data generated through this study is equally significant. It is a valuable resource for educators, policymakers, curriculum designers, and researchers. By making this data accessible, the study contributes to the ongoing efforts to improve the quality and relevance of secondary education in Uganda. Educators can utilize this data to adopt innovative and effective teaching methods that align with the goals of the educational system, ultimately enhancing students’ performance and fostering lifelong learning. Policymakers and curriculum designers can use the insights derived from this data to conduct essential reviews and make informed decisions about teacher competence and the adoption of innovative teaching methodologies. Furthermore, researchers in similar fields can leverage this data to understand better the impact of PBL and the use of multimedia resources in education. This data identifies gaps and challenges and offers potential solutions and avenues for further research.

This data-sharing article presents insights into the effects of Problem-Based Learning (PBL) supplemented by YouTube videos on students’ comprehension of simple machines in physics within Ugandan lower secondary schools. The research collected data from 973 students, encompassing both public and private schools in the Sheema district of Uganda. Three primary types of data were collected: students’ performance data, attitude data, and classroom observation data.

Performance data was acquired through a Physics Learning Achievement Test (PLAT), involving students from various school types and teaching methods. Attitude data were collected via two surveys, one focusing on problem-solving ability (AAPS) and the other on critical thinking ability (CTMS) under PBL with YouTube videos. The Approaches to Problem-Solving Survey (AAPS) and the Critical Thinking Motivational Scale (CTMS) are measurement tools commonly used in the field of physics. The AAPS assesses various strategies individuals employ when solving problems, while the CTMS evaluates motivational factors influencing critical thinking abilities. The Reformed Teaching Observation Protocol (RTOP) assessed classroom practices and teaching methods.

The dataset, available in raw, filtered, and analyzed formats, offers valuable insights into the impact of innovative teaching methods on student performance, attitudes, and classroom practices. It addresses critical questions about the effectiveness of PBL approaches, with potential implications for science education in Uganda.

This dataset intends to assess the impact of PBL when supplemented with YouTube videos on Ugandan form-2 lower secondary schools in learning simple machines. The following are the research questions:

To what extent do PBL and PBL supplemented with YouTube videos enhance students’ conceptual understanding of simple machines in physics?

What are the problem-solving and critical thinking levels brought by learning with PBL supplemented by YouTube videos?

How is physics teaching reformed when learning simple machines in physics with PBL supplemented by YouTube videos?

Are there differences in students’ academic achievement for school type (government alongside private school)?

Ethics statements

The research project rigorously adhered to ethical standards established by the University of Rwanda College of Education’s (UR-CE) Research and Innovation Unit under the ethical protocol number Ref. 03/DRI-CE/078/EN/gi/2021, dated 30th November 2021. All necessary permissions were obtained systematically and ethically, as outlined in the research project description. Here is a summary of the ethical considerations and recruitment process:

Ethical protocol

The research project adhered to the ethical standards and principles of the UR-CE)‘s Research and Innovation Unit. The protocol number and approval date are explicitly mentioned, demonstrating a formal ethical review.

Permissions from authorities

The Ministry of Education and Sports obtained formal permission to access schools through the Permanent Secretary’s (PS) office. The PS communicated with the Chief Administrative Officer (CAO), District Education Officer (DEO), and Resident District Commissioner (RDC) to secure the necessary support for the study.

Engagement with schools

With the approval from the CAO, the DEO contacted school heads to inform them about the research study. The school heads responded positively and even provided physics teachers with three-day problem-based learning (PBL) training as part of the research. It is worth noting that all participating teachers held teaching qualifications, and as part of the research process, we provided them with a three-day training session specifically focused on implementing PBL interventions. This training aimed to ensure consistent delivery of PBL across treatment classrooms and schools, thereby mitigating variations attributable to individual teaching styles.

Informed consent

Teachers and students, with parental consent, willingly participated in the research study. Informed consent forms were signed, indicating they fully understood the study’s purpose, procedures, potential risks, and benefits. Anonymity was ensured for students by not including their names on the test papers.

The research employed purposive sampling to select 973 students from 12 schools. These schools were divided into three groups, each with a different teaching method: PBL with YouTube videos, PBL alone, and Traditional teaching.

Geographic considerations

Schools were selected from different town councils at extreme ends of the district, sharing similar characteristics suitable for the study. This approach helps ensure that the study’s findings are robust and generalizable.

Research design

The study utilized a non-equivalent comparison group pre/post-test design (Creswell, 2012). The study involves Form 2 students from six Sheema District, Western Uganda schools. Three selected schools were public, while the remaining three were private, offering a diverse representation of school types in the district. The selection of schools was purposeful, aiming to ensure diverse representation and maximize the study’s validity. This approach allowed for the strategic allocation of schools to treatment or control groups based on specific criteria pertinent to the research objectives. Notably, the selection criteria considered factors such as geographical location, school size, academic performance, and availability of resources to ensure a balanced representation of different educational contexts. The traditional method, characterized by conventional lectures supplemented with textbooks and teacher-centered content delivery, was employed in control group schools. Students in this group primarily learned through note-taking with minimal demonstrations. Conversely, four other secondary schools were designated as the first treatment group, where Problem-Based Learning (PBL) was implemented. Four additional schools comprised the second treatment group, which utilized PBL supplemented by educational YouTube videos. These groups collectively engaged in constructing knowledge and enhancing conceptual understanding. The participants in the study were form-2 students, ranging in age from 12 to 15 years, who were already enrolled in the schools.

We provide a performance (achievement) test to all 973 students before and after teaching interventions in all groups. We administered an attitude survey (motivation scale) and observed classes in the group that used PBL and YouTube videos. Table 1 presents the sample size under the teaching intervention of design groups implemented.

The objective of the performance test was to gauge students’ grasp of conceptual understanding acquired through the implementation of a problem-based learning approach following the completion of the topic on simple machines. The test, spanning 25 minutes, consisted of ten questions sourced from practice exercises on simple machines within form-two secondary learners’ physics textbooks. The National Curriculum Development Center and the Ministry of Education and Sports in Uganda approved these textbooks. The examination encompassed themes outlined in the approved lower secondary curriculum physics syllabus, covering concepts like the applications of simple machines, mechanical advantage, velocity ratio, and efficiency of machines. Specific topics included levers (covering classes and applications), pulley systems (encompassing types, applications, mechanical advantage, velocity ratio, and efficiency), inclined planes (including applications, mechanical advantage, velocity ratio, and efficiency), wheel and axle (exploring understanding, applications, and velocity ratio), gears (addressing simplification of work, applications, and velocity ratio), and methods of enhancing machine efficiency. The test was validated by four researchers from Mbarara University of Science and Technology (MUST) and the University of Rwanda College of Education (URCE). Test 1 was scored in MS Excel with “IF EXCACT” function, while Test 2 was manually marked, and results were entered in the same software.

Attitude surveys were all adopted from existing literature. Critical Thinking Motivational Scale (CTMS) was used as our Survey 1 and was adapted from Valenzuela et al . 11 , while Attitudes and Approaches to Problem-Solving Survey (AAPS) was used as our Survey 2 and was adapted from Singh and Mason 12 and available at Physport ( https://www.physport.org/assessments/assessment.cfm?A=AAPS ). Problem-solving and critical thinking are integral to effective physics education. They deepen students’ understanding by connecting theoretical concepts to real-world situations 13 , 14 . These skills encourage active engagement and foster analytical abilities, allowing students to break down complex problems. Additionally, they promote creativity, help apply theory to practice, and cultivate logical reasoning. Problem-solving and critical thinking prepare students for future challenges in scientific and engineering fields, encourage collaboration, boost confidence, and instill a mindset for lifelong learning. Incorporating these skills into physics teaching enhances academic performance and equips students with valuable personal and professional growth tools. We adopted all 19 items from CTMS and only 31 items from AAPS to meet our research aim. Thus, the last two items (32 and 33) in AAPS were removed as they were not related to the content delivered in our study. All these surveys were rated on a Likert scale (from strongly disagree to strongly agree). Items 1–4 are related to expectancy, items 5-8 to attainment, items 9-12 to utility, items 13-16 to interest, and items 17-19 to cost.

Classroom observation data was collected with the famous standardized reformed teaching observation protocol (RTOP) from Pibun and Sawada 15 and is available at Pysport ( https://www.physport.org/assessments/assessment.cfm?A=RTOP ). RTOP proved its validity and reliable results across the globe 16 , 17 , 18 , 19 with its potential to reveal reformed teaching while implementing a new teaching method. It comprises 25 statements where each item is evaluated on a 5-scale. It is scored 0 when such practice was not found in a lesson and 4 when a certain practice was very well described or observed in a delivered lesson. During classroom observation, an observer sits in the classroom and observes what the teacher and student do. He/she may take notes on what is happening but wait until the class is over to rate these 25 items.

Data Records

All data described in this descriptor are deposited in figshare ( https://figshare.com/articles/dataset/RTOP_Data_for_the_implementation_of_Problem-based_learning_in_a_Physics_classroom_Uganda/23974902 ) 20

To evaluate the impact of PBL teaching intervention on students’ performance and attitude toward learning physics, we gathered three data types (performance, attitude, and observation) presented in five datasets (two performance tests, two attitude surveys, and one classroom observation).

Students’ performance data

The student performance data comprises two datasets or MS Excel files. The first file contains data for test one titled “Performance data _ Test 1 (Multiple choice) _ 12102022 figshare.” This file contains data from ten multiple-choice questions. The file contains three sheets. The first sheet shows test items (all ten questions), the second presents pretest answer choices, and the third presents post-test answer choices or results. Each results sheet shows the school code (column B), student code (column C), school type (column D), and treatment group (column E) as variables. From column “F” to column “O” we see student answer choices under each test question. From column “Q” to column “Z” we marked the test (one score for each correct question). Column “AB” shows the percent score. Row “3” shows the expected correct answer, while row “4” shows variables and the number of test items.

The second file contains data for test two titled “Performance data _ Test 2 (Problem solving) _ 12102022 figshare.” This file contains data from ten-word problem kinds of questions. The file contains three sheets. The first sheet shows test items (all ten questions), the second presents pretest scores, and the third presents post-test scores or results. Each results sheet shows the school code (column C), student code (column D), school type (column E), and treatment group (column F) as variables. From column “G” to column “P” we see student scores under each test question. Column “R” shows the total score, while column “S” shows percent score. Row “3” shows the assigned score when each question’s expected correct answer was provided. Row “4” shows variables and several test items.

Students’ attitude data

The student attitude data comprises two datasets or MS Excel files. The first file contains data for the first survey titled “Motivation data _ Survey 1 (Critical thinking ability) _ 12102022 figshare.” This file contains data from 19 items of critical thinking ability survey. The file contains two sheets. The first sheet shows the pre-test results, while the second shows the post-test results. Each sheet shows the school code (column C), student code (column D), school type (column E), and treatment group (column F) as variables. From column “F” to column “O” we see student answer choices under each test question. From column “G” to column “Y” we see student answers or agreement (1: STRONGLY DISAGREE, 2: DISAGREE, 3,: NEUTRAL, 4: AGREE, AND 5: STRONGLY AGREE) to each item of the survey. Row “2” shows the survey title, while row “4” shows the variables and number of survey items.

The second file contains data for the second survey titled “Attitude data _ Survey 2 (Problem solving ability) _ 12102022 figshare.” This file contains data from 31 items related to problem-solving ability in learning physics. The file contains two sheets. The first sheet shows the pre-test results, while the second shows post-test results. Each sheet shows the school code (column C), student code (column D), school type (column E), and treatment group (column F) as variables. From column “G” to column “AK” we see student answers or agreement (1: STRONGLY DISAGREE, 2: DISAGREE, 3,: NEUTRAL, 4: AGREE, AND 5: STRONGLY AGREE) to each item of the survey. Row “2” shows the survey title, while row “4” shows the variables and number of survey items.

Classroom observation data

The file for classroom observation data is titled “Classroom observation data _ RTOP for video & pbl group _ 12102022 figshare” and contains only one sheet. From column “B” to column “C” we see RTOP while the following columns (D-AA) present data. Row “10” shows school codes, while row “5” shows several observations and frequencies under each school supplied with PBL and YouTube videos teaching intervention. The data range from 0 (never occurred) to 4 (very descriptive).

Technical Validation

Initially, we had 20 problem-solving questions, but evaluators rated 10 as valid, which were included in the final administration. We also initially had 15 multiple-choice questions, and evaluators rated 10 as appropriate and aligned with the study objectives. A pilot study was conducted with 90 students to evaluate the face validity and reliability of the questions. We assessed the reliability of these items using a split-half method and obtained a high reliability (r = 0.87) for multiple-choice items and a medium reliability (r = 0.68) measured by the Pearson product-moment correlation coefficient for problem-solving items. The split-half reliability assumes that the two halves of the test are equivalent in difficulty and content 21 .

CTMS and AAPS

During our pilot phase, the internal consistency of CTMS, assessed using Cronbach’s alpha, was found to be high (0.793) for all 19 items, medium (0.428) for expectancy, (0.411) for attainment, (0.686) for utility, (0.574) for interest, and (0.594) for cost. The AAPS exhibited an internal consistency reliability of Cronbach’s alpha = 0.685. It is important to note that the AAPS contains nine items formulated negatively. Therefore, for a positive attitude, students were required to respond with ‘Disagree’ or ‘Strongly disagree’ to these items (1, 3, 5, 8, 11, 12, 16, 23, and 30). Consequently, the reliability of the 22 positively formulated items was 0.601, while that of the negatively formulated items was 0.480.

Before observing actual classes, we underwent a 2-hour training session and watched and coded a YouTube classroom video on physics. The inter-rater agreement between the first author and the assistant exceeded 80% on two occasions, indicating the reliability of the data.

Scope and potential limitations

In our study, we recognized the significance of investigating potential bias in the results obtained from students in both private and public schools. To ensure the credibility and robustness of our findings, we conducted a comparative analysis to determine whether any notable disparities existed between these two groups. Our data collection process was comprehensive, encompassing a diverse range of schools, including both private and public institutions. This approach allowed us to capture a broad spectrum of socioeconomic backgrounds and educational settings. The study itself involved Form-2 students who were enrolled in schools situated in different town councils at opposite ends of the district. Despite their geographical diversity, these schools shared pertinent characteristics relevant to our research objectives. To facilitate our investigation, we categorized these schools into distinct treatment groups, comprising PBL alone and PBL with videos, along with a control group following traditional teaching methods. Importantly, we deliberately chose to maintain the existing class arrangements in these schools. Our commitment to preserving each school’s established class organization and cultural norms guided this decision.

However, it is essential to acknowledge the limitations inherent in the research design. One notable limitation is the observation of attitudes, which was limited to the student group exposed to the PBL with the video teaching method. This restriction may impact the generalizability of the findings, as attitudes toward learning may vary among students exposed to different instructional methods. Future research endeavors could consider incorporating measures to assess attitudes across all treatment groups to provide a more comprehensive understanding of the intervention’s effects.

The current data files do not contain information on individual teachers due to the scope and focus of the study. These variables could include educators’ teaching experience, pedagogical approach, content knowledge, and instructional effectiveness. Since we recognize the significance of teacher impact, we would consider incorporating such variables in future research projects to provide a more comprehensive analysis of instructional effectiveness and its associated factors.

Regarding the decision to maintain existing class arrangements in schools, particularly considering cultural norms, it is crucial to recognize its potential influence on the study outcomes. The intervention’s impact may have been influenced by preserving the existing class structures, including student composition and dynamics. For instance, certain class arrangements may foster greater collaboration and engagement, while others may present challenges in implementing collaborative learning approaches such as PBL. Therefore, future studies could explore the relationship between class arrangements and instructional effectiveness to provide insights into optimizing learning environments.

Usage Notes

Value of the data.

The data presented is valuable and beneficial to science education in Uganda as it elucidates the status of students’ content knowledge and their perceptions about learning simple machines with PBL approaches.

Policymakers and curriculum designers have the opportunity to conduct essential reviews that highlight the competence of teachers. This process can pave the way for advocating innovative and relevant teaching methodologies, subsequently informing the identification of professional development requirements for educators.

Researchers in similar fields can re-use these data to measure the effect of PBL intervention on student achievement, identify gaps, and predict possible remedies. Thus, data can be analyzed using various variables such as teaching intervention and school type.

Code availability

No custom code was used.

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Acknowledgements

We acknowledge the African Center for Excellence for Innovative Teaching and Learning Mathematics and Science (ACEITLMS) for funding this study and the authors of the research tools we used to free them to use. All study participants, teachers, and school headteachers are also well acknowledged.

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Nicholus Gumisirizah: Conceptualization, Methodology, Visualization, Data curation, Software, Writing- Original draft preparation. Joseph Nzabahimana, Charles M. Muwonge: Conceptualization, Validation, Writing- Reviewing and Editing.

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Correspondence to Nicholus Gumisirizah .

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Gumisirizah, N., Nzabahimana, J. & Muwonge, C.M. Students’ performance, attitude, and classroom observation data to assess the effect of problem-based learning approach supplemented by YouTube videos in Ugandan classroom. Sci Data 11 , 428 (2024). https://doi.org/10.1038/s41597-024-03206-2

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Activity based learning – Summary

Activity-based learning is a teaching approach that focuses on engaging students in hands-on activities to enhance their learning experience. It involves interactive tasks, group activities, and real-world applications to promote critical thinking and problem-solving skills. By incorporating various resources and encouraging creativity, activity-based learning aims to make learning more dynamic and meaningful for students.

Here are some examples that personifies activity-based learning:

1. Science Experiments: Conducting experiments in the science lab to explore scientific concepts hands-on.

2. Group Projects: Working in teams to research, collaborate, and present findings on a specific topic.

3. Role-Playing Simulations: Acting out historical events or real-life scenarios to understand different perspectives.

4. Field Trips: Visiting museums, nature reserves, or historical sites to experience learning outside the classroom.

Written by INSHAASHRAF

Active learning

Class room management.

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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 ...
Full article: Understanding and explaining pedagogical problem solving
1. Introduction. The focus of this paper is on understanding and explaining pedagogical problem solving. This theoretical paper builds on two previous studies (Riordan, Citation 2020; and Riordan, Hardman and Cumbers, Citation 2021) by introducing an 'extended Pedagogy Analysis Framework' and a 'Pedagogical Problem Typology' illustrating both with examples from video-based analysis of ...
(PDF) Principles for Teaching Problem Solving
structured problem solving. 7) Use inductive teaching strategies to encourage synthesis of mental models and for. moderately and ill-structured problem solving. 8) Within a problem exercise, help ...
Teaching Mathematics Through Problem Solving
Teaching about problem solving begins with suggested strategies to solve a problem. For example, "draw a picture," "make a table," etc. You may see posters in teachers' classrooms of the "Problem Solving Method" such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no ...
The process of implementing problem-based learning in a teacher
Using group discussions, the group members discussed issues related to the situation by presenting the problem and leading the whole class in analysing/discussing and summarising the problem and developing/organising the appropriate teaching approach. In this way, the teaching content developed gradually as a group effort.
Problem-Based Learning (PBL)
PBL is a student-centered approach to learning that involves groups of students working to solve a real-world problem, quite different from the direct teaching method of a teacher presenting facts and concepts about a specific subject to a classroom of students. Through PBL, students not only strengthen their teamwork, communication, and ...
Teaching: the problem-solving approach
Teaching: the problem-solving approach - UNESCO ... article
Learning to Teach Mathematics Through Problem Solving
Teaching and learning mathematics through problem solving supports learners' development of deep and conceptual understandings (Inoue et al., 2019 ), and is regarded as an effective way of catering for diversity (Hunter et al., 2018 ). While the importance and challenge of mathematical problem solving in school classrooms is not questioned ...
PDF Problem Based Learning: A Student-Centered Approach
principles and concept. PBL is both a teaching method and approach to the curriculum. It can develop critical thinking skill, problem solving abilities, communication skills and lifelong learning. The purpose of this study is to give the general idea of PBL in the context of language learning, as PBL has expanded in the areas of law,
PDF A Problem With Problem Solving: Teaching Thinking Without Teaching ...
Problem Solving Teaches Creativity Real world situations require creativity. However, it has often been claimed that traditional classrooms or teaching approaches do not focus on developing the creative faculty of students. Advocates of problem solving, by contrast, claim that problem solving
PDF Problem-Based Teaching of Literature
problem-situation, problem solving, problem-solving teaching, and problem-solving learning. Nevertheless, in contemporary teaching methodology, the problem-based creative methodology system is the most widespread under the term problem-based teaching (Gajić, 2004: 139). In changing the organization of work and approach to teaching
Elementary teachers' experience of engaging with Teaching Through
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 ...
Problem-Solving Method In Teaching
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 ...
(PDF) Enhancing students' problem-solving skills ...
The problem solving approach to teaching technology education. content closely parallels the way technologists think and go about the task. of finding solutions to problems. This approach provides ...
Students' performance, attitude, and classroom observation data to
The Approaches to Problem-Solving Survey (AAPS) and the Critical Thinking Motivational Scale (CTMS) are measurement tools commonly used in the field of physics.
Activity based learning
Activity-based learning is a teaching approach that focuses on engaging students in hands-on activities to enhance their learning experience. It involves interactive tasks, group activities, and real-world applications to promote critical thinking and problem-solving skills.
Problem Solving for the Future of Legal Work Through Design Thinking
and into a problem-solving approach that continues to ask even better questions and knows how to apply the answers. It starts with empathy and curiosity toward the needs of lawyers and clients to imagine (and regularly re-imagine) the law firm of the future. In practice, it looks like asking attorneys some defining questions. ...

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5 Teaching Mathematics Through Problem Solving

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

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Low Floor High Ceiling Tasks

Math in 3-Acts

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

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Lesson Study and Its Role in the Implementation of Curriculum Reform in China

Learning to Teach Mathematics Through Problem Solving

Introduction

Teaching Through Problem Solving (TTP)

Demands of TTP

Supports for teachers

Teachers’ reservations about TTP

Problem solving practices in Irish elementary mathematics education

Lesson Study (LS) as a professional development model

Context of this study

Research questions

Methodology

LS phase 1: Study

LS phase 2: Planning

LS phase 3: Implementation

LS phase 4: Reflection