discuss the problem solving method of teaching

Problem-Solving Method in Teaching

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

Table of Contents

Definition of problem-solving method.

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

Meaning of Problem-Solving Method

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

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

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

Benefits of Problem-Solving Method

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

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

Steps in Problem-Solving Method

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

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

Verification of the concluded solution or Hypothesis

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

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

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

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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 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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5 Teaching Mathematics Through Problem Solving

Janet Stramel

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

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

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

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

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

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

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

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

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

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

Key features of a good mathematics problem includes:

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

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

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

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

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

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

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

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

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

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

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

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

Number Talks

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

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

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

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

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

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

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

Instead, you and your students could say the following:

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

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

Using “Worksheets”

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

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

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

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

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

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

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

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

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

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

when we teach students how to problem solve

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

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

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

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

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• Faculty & Staff

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.

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. 

The Harriet W. Sheridan Center for Teaching and Learning

Teaching problem solving.

• Teaching Resources
• Course Design

The other day, a physicist friend was working in the lab with her summer research students. They were talking about the work they’d been doing that summer and how there was no manual or instructions of any sort for any of it; no textbook, no lab procedure. It was as if they were making it up as they went along. Laughing about this, one of the students said, ‘You know what we need? We need an entire course with nothing but problems. Just give us one problem after another, and we figure out how to do them. Because that’s what real research is.’ The rest of the students laughed. And then all of them nodded.

Employers, college presidents, faculty, and students demonstrate remarkable consensus that problem solving is one of the most important outcomes of a college education (Bok, 2017; Hart Research Associates, 2015; Hora, Benbow, Oleson, 2016; Passow & Passow, 2017). At the time of this newsletter, there were 28 courses offered this year that included the words “problem*” and “solving” in Courses@Brown. Course descriptions ranged from focusing on how to apply techniques or skills, to solving problems, to tackling common problems encountered in the field, and concepts that included “problems” within their title. There are undoubtedly more courses that implicitly and explicitly focus on problem solving across campus. In light of this emphasis, it is important to ask, “What is a problem and what is problem solving?” and “How do I foster problem-solving skills in my course?” and eventually, "How will I be explicit about problem solving in my course and course description?" Although problem solving is often associated with STEM courses, this newsletter offers perspectives and teaching approaches from across the disciplines.

What is a “problem” and problem solving?

Problems and problem solving may be context and discipline specific, but the concept and process have overarching components and similarities across contexts. Jonassen (2000, p. 65) defines a problem as an “unknown entity in some situation (the difference between a goal state and a current state)” such that “finding or solving for the unknown must have some social, cultural, or intellectual value.”  Within courses, students may encounter a wide variety of current (e.g., a problem statement) and goal (e.g., a solution) states with different motivations for solving them. Students will be exposed to “well-structured” problems at one end of the spectrum, which have a typical solution path and solution, and “ill-structured” problems, which are highly context dependent and have no one solution path (Jonassen, 2000).

We bring in common case scenarios for students and try to develop the frameworks they need to approach a problem rather than just finding the answer. To help students think about the process, we scaffold scenarios over the years through self-study modules that students can complete on their own. The scenarios stay the same, but students can come back to them with new information and frameworks they have learned, a deeper toolbox to pull from in different clinical settings. This allows students to be lifelong learners and more flexible and adaptable in the future.

Problem solving is a “goal-oriented” process that includes creating and manipulating problems as mental models (Jonassen, 2000). Brown faculty from a variety of disciplines were interviewed by Sheridan staff and asked, “What skills do students need to problem solve effectively?” They responded that students need to be able to do the following:

• Reason, observe, and recognize patterns
• Use current information to understand the past
• Know how to break complex problems down into smaller, more manageable components
• Make connections between concepts and disciplines
• Creatively think of multiple solution paths

These skills, among others, target the following problem-solving steps (Pretz, Naples, & Sternbergy, 2003):

• Recognize or identify a problem
• Define and represent the problem mentally
• Develop a solution strategy
• Organize your knowledge about the problem
• Allocate mental and physical resources for solving the problem
• Monitor your progress toward the goal
• Evaluate the solution for accuracy

Problem solving is an iterative process, and as such, these steps do not necessarily progress in a linear fashion. When creating homework assignments, projects, exams, etc., it is helpful to identify the specific skills you want students to practice, the strategies they should use, and how you will evaluate the solutions they produce.

How do I foster problem-solving skills in my course?

Instructors can signpost the problem-solving skills students should develop in their courses by adapting existing problem sets to fit recommendations from the  Transparency in Learning and Teaching Project  (TILT). The process of increasing transparency in assignments includes communicating the assignment’s purpose, task, and criteria to students (Winkelmes et al., 2016):

• The purpose usually links to one learning objective for the course, the skills students will develop as a result of completing the assignment, or a real-world application that students might experience outside of your classroom. In this way, the problem you have presented to the student becomes relevant because it has “some social, cultural, or intellectual value” (Jonassen, 2000, p. 65).
• Next, the task states the strategy or strategies students should take to complete the assignment. This includes guiding students through organizing the information available to develop a strategy.
• Finally, the criteria could be a rubric or annotated examples that are given to students  before  the assignment is due, so they are aware of the standards for the assignment.

In one study, researchers found that in courses where at least two assignments had features of transparent assignments, students self reported increases in their academic confidence, sense of belonging, and mastery of skills, such as problem solving (Winkelmes et al., 2016). Below are examples of different skills needed for problem solving with suggestions on how you can foster these skills through adapted or new assignments and in-class exercises.

A key skill for problem solving is knowing how to define and represent the problem and its solutions. This is true for all students, regardless of discipline. For example, Berkenkotter (1982, p. 33) states, “A writer is a problem solver of a particular kind. Writers’ ‘solutions’ will be determined by how they frame their problems, the goals they set for themselves, and the means or plans they adopt for achieving those goals.” To help students understand and connect to research in their field, instructors can assign short articles and guide students through rhetorical practices to make expert thinking more explicit. Provide students multiple opportunities to refine their writing allows them to learn “how to frame their problems.”

The distant past can seem uncomfortably strange to modern observers. As we discuss our class readings, one thing I like to do with my students is to explore what seems weird or even offensive to them about our texts and the societies that produced them. Thinking about the disconnect between ancient and modern attitudes, outlooks, beliefs, and values can be an incredibly productive way to think about cultural difference over space and time. - Professor Jonathan Conant, History and Classics

Critical thinking is the “ability to assess your assumptions, beliefs, and actions” (Merriam & Bierema, 2014, p. 222) with the intent to change your actions in the future and is necessary when solving problems. It is a skill required during all steps of the problem-solving process. Fostering critical thinking in your students is one way to create a more inclusive classroom because you are inherently asking students to challenge their assumptions and biases.

Instructors can use the following conditions to promote critical thinking in your classroom (Merriam & Bierema, 2014):

• Foster critical reflection by examining assumptions (see  Promoting Metacognition  for specific reflective strategies), e.g., ask students to read a research article and identify possible assumptions that are made in the questions asked, methods used, or the interpretation of the results. For example, to foster critical reflection you could ask students to identify the sources of knowledge they value and use when completing homework and write a reflection on what assumptions they made about those sources. What are the identities of the people creating those sources of knowledge? What systems or people are gatekeepers of that knowledge?
• Build a learning community where the expectation is that students can be wholly present, honest, ask questions, and productively fail (Kapur, 2016).
• Practice dialogical conversation by teaching an awareness of power relations in the classroom such as microaggressions or micro-affirmations and how to use active listening (see  Microaggressions and Micro-aggressions  for examples and specific practices).
• Provide students the opportunity to make connections between content and their experiences, e.g., by asking students on homework assignments how they apply concepts to a recent experience or asking students why they took your course and how it relates to their career goals.

Instructors can develop aspects of problem solving by being intentional about team building, connecting students to alternative perspectives, and being explicit about the expectations of teamwork in the field (e.g., as a researcher, industry partner, consultant, etc.). You can create homework assignments using the  TILT framework , which asks students to evaluate both their own and peers’ interactions in teams. There are several models or rubrics for how to assess teamwork, such as the  AAC&U Teamwork Value Rubric , which focuses on students’ behaviors or the  Comprehensive Assessment of Team Member Effectiveness (CATME) , which is a free packaged tool that gathers information from students and groups them into teams.

We use team-based learning exercises and collaborative problem solving. Students are assigned pre-reading to expand their knowledge so they are able to think through different aspects of a scenario before they come to class. In class, the discussion focuses on a team deciding and agreeing on what the next steps in a case will be. The problem-solving skills that this team discussion focuses on are interpersonal communication, being an active listener, and a collaborative team member. It is not high stakes, but together the team will succeed or fail. - Sarita Warrier, Assistant Dean for Medical Education, Alpert Medical School

A  jigsaw  is another collaborative approach to teach students how to break up a problem into smaller components. For example, in a class on Romanticism and Romantic philosophies, three groups of students would each be given the following questions about five poems: “How does the writer view nature?” (Group 1), “How does the writer view society?” (Group 2), “How does the writer view the purpose of poetry?” (Group 3). After discussion, three new groups, with representatives from each of these three clusters, might discuss a broader question, such as, “Using the information gathered in the first groups [...] what are Romanticism’s goals? What’s the agenda of the Romantic poets?” (Handstedt, 2018, pp. 121-122).

Expert researchers, practitioners, and educators incorporate reflection and iteration as part of their practice. Key steps of the problem-solving process include being reflective about the process and what is working or not working towards a goal. In a previous newsletter,  Promoting Metacognition , the Sheridan Center provided a list of several activities and assignments you could use to help students be reflective in your course. These activities range from short  minute papers , to semester-long reflective journals. Think-alouds, or having a student verbally solve a problem with another student, can also help students develop reflective problem-solving skills because it “provides a structure for students to observe both their own and another’s process of learning” (Barkley, 2010, p. 259).

For more strategies on how to engage students in these skills and topics, please see the Sheridan Center’s newsletter,  Inclusive Teaching Through Active Learning . It is important to be explicit in how you approach problem solving and convey that information both through your course description, syllabi, and content.

Opportunities at Sheridan for Development of Problem Solving

Problem solving is a necessary skill in all disciplines and one that the Sheridan Center is focusing on as part of the  Brown Learning Collaborative , which provides students the opportunity to achieve new levels of excellence in six key skills traditionally honed in a liberal arts education ­– critical reading, writing, research, data analysis, oral communication, and problem solving. To help you think through how to integrate opportunities for students to problem solve effectively in your course, the Sheridan Center offers problem solving professional development opportunities for faculty and students in an effort to engage intergenerational, faculty-student teaching teams.

Problem-Solving Course Design Institute

Problem-solving fellows program.

Berkenkotter, C. (1982). Writing and problem solving. In T. Fulwiler & A. Young (Eds.),  Language connections: Writing and reading across the curriculum  (pp. 33-44). Urbana, Illinois: National Council of Teachers of English.

Barkley, E.F. (2010).  Student engagement techniques: A handbook for college faculty . San Francisco, CA: Jossey-Bass.

Bok, D. (2017).  The struggle to reform our colleges.  Princeton, NJ: Princeton University Press.

Hanstedt, P. (2018).  Creating wicked students: Designing courses for a complex world . Sterling, VA: Stylus.

Hart Research Associates. (2015).  Falling short? College learning and career success . Survey carried out for AAC&U. Available: https://www.aacu.org/sites/default/files/files/LEAP/2015employerstudents…

Hora, M.T., Benbow, R. J., & Oleson, A. K.. (2016).  Beyond the skills gap: Preparing college students for life and work . Cambridge, MA: Harvard University Press.

Jonassen, D. H. (2000). Toward a design theory of problem solving.  Educational technology research and development , 48(4), 63-85.

Kapur, M. (2016). Examining productive failure, productive success, unproductive failure, and unproductive success in learning.  Educational Psychologist , 51(2), 289-299.

Merriam, S. B., & Bierema, L. L. (2014).  Adult learning: Linking theory and practice . John Wiley & Sons.

Passow, H.J., & Passow, C.H. (2017). What competencies should undergraduate engineering programs emphasize? A systematic review.  Journal of Engineering Education , 106(3): 475-526.

Pretz, J.E., Naples, A. J., & Sternbergy, R. J. (2003). Recognizing, defining, and representing problems. In J. E. Davidson & R. J. Sternberg (Eds.),  The psychology of problem solving  (pp. 3-30). New York: Cambridge University Press.​

Winkelmes, M.A., Bernacki, M., Butler, J., Zochowski, M., Golanics, J., & Weavil, K. H. (2016). A teaching intervention that increases underserved college students’ success.  Peer Review , 18(1/2), 31–36.

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6 Tips for Teaching Math Problem-Solving Skills

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

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

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

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

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

1. Link problem-solving to reading

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

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

2. Avoid boxing students into choosing a specific operation

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

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

3. Revisit ‘representation’

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

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

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

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

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

4. Give time to process

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

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

5. Ask questions that let Students do the thinking

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

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

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

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

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

Problem Solving Method Of Teaching

The problem-solving method of teaching is the learning method that allows children to learn by doing. This is because they are given examples and real-world situations so that the theory behind it can be understood better, as well as practice with each new concept or skill taught on top of what was previously learned in class before moving onto another topic at hand.

What is your preferred problem-solving technique?

Answers : - I like to brainstorm and see what works for me - I enjoy the trial and error method - I am a linear thinker

Share it with me by commenting.

For example, while solving a problem, the child may encounter terms he has not studied yet. These will further help him understand their use in context while developing his vocabulary. At the same time, being able to practice math concepts by tapping into daily activities helps an individual retain these skills better.

One way this type of teaching is applied for younger students particularly is through games played during lessons. By allowing them to become comfortable with the concepts taught through these games, they can put their knowledge into use later on. This is done by developing thinking processes that precede an action or behavior. These games can be used by teachers for different subjects including science and language.

For younger students still, the method of teaching using real-life examples helps them understand better. Through this, it becomes easier for them to relate what they learned in school with terms used outside of school settings so that the information sticks better than if all they were given were theoretical definitions. For instance, instead of just studying photosynthesis as part of biology lessons, children are asked to imagine plants growing inside a dark room because there is no sunlight present. When questioned about the plants, children will be able to recall photosynthesis more easily because they were able to see its importance in real life.

Despite being given specific examples, the act of solving problems helps students think for themselves. They learn how to approach situations and predict outcomes based on what they already know about concepts or ideas taught in class including the use of various skills they have acquired over time. These include problem-solving strategies like using drawings when describing a solution or asking advice if they are stuck to unlock solutions that would otherwise go beyond their reach.

Teachers need to point out in advance which method will be used for any particular lesson before having children engage with it. By doing this, individuals can prepare themselves mentally for what is to come. This is especially true for students who have difficulty with a particular subject. In these cases, the teacher can help them get started by providing a worked example for reference or breaking the problem down into manageable chunks that are easier to digest.

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Ultimately, the goal of teaching using a problem-solving method is to give children the opportunity to think for themselves and to be able to do so in different contexts. Doing this helps foster independent learners who can utilize the skills they acquired in school for future endeavors.

The problem-solving method of teaching allows children to learn by doing. This is because they are given examples and real-world situations so that the theory behind it can be understood better, as practice with each new concept or skill taught on top of what was previously learned in class before moving onto another topic at hand.

One way this type of teaching is applied for younger students particularly is through games played during lessons. By allowing them to become comfortable with the concepts taught through these games, they are able to put their knowledge into use later on. This is done by developing thinking processes that precede an action or behavior. These games can be used by teachers for different subjects including science and language.

For instance, a teacher may ask students to imagine they are plants in a dark room because there is no sunlight present. When questioned about the plants, children will be able to recall photosynthesis more easily because they were able to see its importance in real life.

It is important for teachers to point out in advance which method will be used for any particular lesson before having children engage with it. By doing this, individuals can prepare themselves mentally for what is to come. This is especially true for students who have difficulty with a particular subject. In these cases, the teacher can help them get started by providing a worked example for reference or breaking the problem down into manageable chunks that are easier to digest.

lesson before having children engage with it. By doing this, individuals can prepare themselves mentally for what is to come. This is especially true for students who have difficulty with a particular subject. In these cases, the teacher can help them get started by providing a worked example for reference or breaking the problem down into manageable chunks that are easier to digest.

The teacher should have a few different ways to solve the problem.

For example, the teacher can provide a worked example for reference or break down the problem into chunks that are easier to digest.

The goal of teaching using a problem-solving method is to give children the opportunity to think for themselves and to be able to do so in different contexts. Successful problem solving allows children to become comfortable with concepts taught through games that develop thinking processes that precede an action or behavior.

Introduce the problem

The problem solving method of teaching is a popular approach to learning that allows students to understand new concepts by doing. This approach provides students with examples and real-world situations, so they can see how the theory behind a concept or skill works in practice. In addition, students are given practice with each new concept or skill taught, before moving on to the next topic. This helps them learn and retain the information better.

Explain why the problem solving method of teaching is effective.

The problem solving method of teaching is effective because it allows students to learn by doing. This means they can see how the theory behind a concept or skill works in practice, which helps them understand and remember the information better. This would not be possible if they are only told about the new concept or skill, or read a textbook to learn on their own. Since students can see how the theory works in practice through examples and real-world situations, the information is easier for them to understand.

List some advantages of using the problem solving method of teaching.

Some advantages of using the problem solving method of teaching are that it helps students retain information better since they are able to practice with each new concept or skill taught until they master it before moving on to another topic. This also allows them to learn by doing so they will have hands-on experience with facts which helps them remember important facts faster rather than just hearing about it or reading about it on their own. Furthermore, this teaching method is beneficial for students of all ages and can be adapted to different subjects making it an approach that is versatile and easily used in a classroom setting. Lastly, the problem solving method of teaching presents new information in a way that is easy to understand so students are not overwhelmed with complex material.

The problem solving method of teaching is an effective way for students to learn new concepts and skills. By providing them with examples and real-world situations, they can see how the theory behind a concept or skill works in practice. In addition, students are given practice with each new concept or skill taught, before moving on to the next topic. This them learn and retain the information better.

What has been your experience with adopting a problem-solving teaching method?

How do you feel the usefulness of your lesson plans changed since adopting this method?

What was one of your most successful attempts in using this technique to teach students, and why do you believe it was so successful?

Were there any obstacles when trying to incorporate this technique into your class? 

Did it take a while for all students to get used to the new type of teaching style before they felt comfortable enough to participate in discussions and ask questions about their newly acquired knowledge?

What are your thoughts on this method? 

“I have had the opportunity to work in several districts, including one where they used problem solving for all subjects. I never looked back after that experience--it was exciting and motivating for students and teachers alike." 

"The problem solving method of teaching is great because it makes my subject matter more interesting with hands-on activities."

What is the role of educators in facilitating problem-solving method of teaching?

Role of Educators in Facilitating Problem-Solving Understanding the Problem-Solving Method The problem-solving method of teaching encourages students to actively engage their critical thinking skills to analyze and seek solutions to real-world problems. As such, educators play a crucial part in facilitating this learning style to ensure the effective attainment of desired skills. Encouraging Collaboration and Communication One of the ways educators can facilitate problem-solving is by promoting collaboration and communication among students. Working as a team allows students to share diverse perspectives while considering multiple solutions, thereby fostering an open-minded and inclusive environment that is crucial for effective problem-solving. Creating a Safe Space for Failure Educators must recognize that failure is an integral component of the learning process in a problem-solving method. By establishing a safe environment that allows students to fail without facing judgment or embarrassment, teachers enable students to develop perseverance, resilience, and an enhanced ability to learn from mistakes. Designing Relevant and Engaging Problems The selection and design of appropriate problems contribute significantly to the success of the problem-solving method of teaching. Educators should focus on presenting issues that are relevant, engaging, and age-appropriate, thereby sparking curiosity and interest amongst students, which further improves their problem-solving abilities. Scaffolding Learning Scaffolding is essential in the problem-solving method for providing adequate support when required. Teachers need to break down complex problems into smaller, manageable steps, and gradually remove support as students develop the necessary skills, thus promoting their self-reliance and independent thinking. Providing Constructive Feedback Constructive feedback from educators is invaluable in facilitating the problem-solving method of teaching, as it enables students to reflect on their progress, recognize areas for improvement, and actively develop their critical thinking and problem-solving abilities. In conclusion, the role of educators in facilitating the problem-solving method of teaching comprises promoting collaboration, creating a safe space for failure, designing relevant problems, scaffolding learning, and providing constructive feedback. By integrating these elements, educators can help students develop essential life-long skills and effectively navigate the complex world they will experience.

Can interdisciplinary approaches be incorporated into problem-solving teaching methods, and if so, how?

Interdisciplinary Approaches in Problem-Solving Teaching Methods Integration of Interdisciplinary Approaches Incorporating interdisciplinary approaches into problem-solving teaching methods can be achieved by integrating various subject areas when presenting complex problems that require students to draw from different fields of knowledge. By doing so, learners will develop a deeper understanding of the interconnectedness of various disciplines and improve their problem-solving skills. Project-Based Learning Activities Implementing project-based learning activities in the classroom allows students to work collaboratively on real-world problems. By involving learners in tasks that necessitate the integration of diverse subjects, they develop the ability to transfer skills acquired in one context to novel situations, thereby expanding their problem-solving abilities. Role of Teachers in Interdisciplinary Teaching Teachers play a crucial role in the successful incorporation of interdisciplinary methods in problem-solving teaching. They must be prepared to facilitate student-centered learning and engage in ongoing professional development tailored towards interdisciplinary education. In doing so, educators can create inclusive learning environments that encourage individualized discovery and the application of diverse perspectives to solve complex problems. Benefits of Interdisciplinary Teaching Methods Adopting interdisciplinary teaching methods in problem-solving education not only enhances students' problem-solving abilities but also fosters the development of critical thinking, creativity, and collaboration. These essential skills enable learners to navigate and adapt to an increasingly interconnected world and have been shown to contribute to students' academic and professional success. In conclusion, incorporating interdisciplinary approaches into problem-solving teaching methods can be achieved through the integration of various subject areas, implementing project-based learning activities, and the active role of teachers in interdisciplinary education. These methods benefit students by developing problem-solving skills, critical thinking, creativity, and collaboration, preparing them for future success in an interconnected world.

In what ways can technology be integrated into the problem-solving method of instruction?

**Role of Technology in Problem-Solving Instruction** Technology can be integrated into the problem-solving method of instruction by enhancing student engagement, promoting collaboration, and supporting personalized learning. **Enhancing Student Engagement** One way technology supports the problem-solving method is by increasing students' interest through interactive and dynamic tools. For instance, digital simulations and educational games can help students develop critical thinking and problem-solving skills in a fun, engaging manner. These tools provide real-world contexts and immediate feedback, allowing students to experiment, take risks, and learn from their mistakes. **Promoting Collaboration** Technology also promotes collaboration among students, as online platforms facilitate communication and cooperation. Utilizing tools like video conferencing and shared workspaces, students can collaborate on group projects, discuss ideas, and solve problems together. This collaborative approach fosters a sense of community, mutual support, and collective problem-solving. Moreover, it helps students develop essential interpersonal skills, such as teamwork and communication, which are crucial in today's workplaces. **Supporting Personalized Learning** Finally, technology can be used to provide personalized learning experiences tailored to individual learners' needs, interests, and abilities. With access to adaptive learning platforms or online resources, students can progress at their own pace, focus on areas where they need improvement, and explore topics that interest them. This kind of personalized approach allows instructors to identify areas where students struggle and offer targeted support, enhancing the problem-solving learning experience. In conclusion, integrating technology into the problem-solving method of instruction can improve the learning process in various ways. By fostering student engagement, promoting collaboration, and facilitating personalized learning experiences, technology can be employed as a valuable resource to develop students' problem-solving skills effectively.

I graduated from the Family and Consumption Sciences Department at Hacettepe University. I hold certificates in blogging and personnel management. I have a Master's degree in English and have lived in the US for three years.

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Heuristic Method Of Teaching: A Guide for Teachers

The heuristic method is a student-centered approach that actively encourages learners to explore and discover knowledge through problem-solving and inquiry-based learning.

In this article, I will delve into the origin and development of the heuristic method, explain its principles and steps, discuss its advantages and disadvantages, provide examples of successful implementation, and explore ways to incorporate it into different subjects.

Table of Contents

The origin and development of the heuristic method of teaching

The renowned mathematician George Pólya developed the heuristic teaching method in the early 20th century . Pólya believed students should be taught how to think rather than what to think.

He recognized the importance of problem-solving and inquiry as effective ways to deepen understanding and foster creativity.

Pólya’s groundbreaking book, “How to Solve It,” outlined a four-step problem-solving process that became the foundation of the heuristic method.

Understanding the heuristic method of teaching

What exactly is the heuristic method of teaching? At its core, it is an approach that encourages students to actively engage in problem-solving, critical thinking, and self-directed learning.

Unlike traditional teaching methods focusing on rote memorization and passive learning, the heuristic method empowers students to take ownership of their learning journey .

Teachers can guide students toward discovering solutions and constructing their knowledge by posing open-ended questions, encouraging exploration, and promoting independent thinking.

Applying the heuristic method of teaching in mathematics

Mathematics is an ideal subject for implementing the heuristic teaching method due to its inherent problem-solving nature. When teaching mathematics using the heuristic method, teachers present students with challenging problems that require them to think critically and apply mathematical concepts and principles.

Instead of providing step-by-step solutions, teachers guide students through problem-solving, encouraging them to explore different strategies, make connections, and justify their reasoning.

This approach enhances students’ mathematical skills and develops their problem-solving abilities, critical thinking, and perseverance.

Steps involved in the heuristic method of teaching

There are several steps that teachers should follow to implement the heuristic method of teaching effectively. Firstly, teachers must present students with open-ended problems that stimulate their curiosity and require them to think critically.

Secondly, teachers should guide students in exploring different problem-solving strategies, encouraging them to brainstorm ideas and consider multiple approaches.

Thirdly, students should be free to experiment, make mistakes, and learn from errors. Fourthly, teachers should facilitate discussions and provide guidance, asking probing questions to deepen students’ understanding and prompt reflection.

Advantages of using the heuristic method of teaching

The heuristic method of teaching offers several advantages over traditional teaching methods. Firstly, it promotes active engagement and student-centered learning, fostering a sense of ownership and autonomy.

Students participate actively in their learning journey, developing problem-solving skills, critical thinking abilities, and creativity.

Secondly, the heuristic method encourages students to think outside the box and explore different perspectives, enhancing their ability to tackle complex problems in various domains.

Thirdly, by focusing on the process rather than the final answer, the heuristic method cultivates a growth mindset and resilience in students, instilling the belief that mistakes are opportunities for learning and growth.

Disadvantages and challenges of the heuristic method of teaching

While the heuristic teaching method has numerous advantages, it has challenges. One of the main difficulties teachers may face is the time-consuming nature of the approach.

Guiding students through the problem-solving process and facilitating discussions can be time-intensive, making it challenging to cover a wide range of topics within a limited timeframe.

Additionally, implementing the heuristic method requires a shift in the traditional teacher-student dynamic, which may require additional training and support for educators.

Finally, assessing students’ understanding and progress can be more complex in a heuristic classroom, as the focus is on the process rather than the final answer.

Examples of successful implementation of the heuristic method of teaching

Numerous educators have successfully implemented the heuristic method of teaching across different subjects.

For example, teachers have used open-ended problems to engage students in authentic mathematical thinking. Students are encouraged to explore multiple strategies, justify their reasoning, and communicate their thought processes.

Similarly, in science, the heuristic method can be applied by posing real-world problems that require students to apply scientific concepts and conduct investigations.

Incorporating the heuristic method of teaching different subjects

While the heuristic method of teaching has been commonly associated with mathematics and science, it can be adapted and incorporated into various subjects.

In literature, for instance, students can explore different interpretations of texts, engage in discussions, and analyze complex themes and characters. In history, students can investigate primary sources, critically evaluate historical events, and draw connections to contemporary issues.

By applying the principles of the heuristic method, teachers can create dynamic and interactive learning experiences that transcend traditional subject boundaries.

Conclusion: Embracing the power of the heuristic method in education

In conclusion, the heuristic method of teaching is a powerful approach that empowers students to become active learners, critical thinkers, and problem solvers.

By embracing this student-centered approach, educators can foster a love for learning, develop essential skills, and cultivate a growth mindset in their students.

While implementing the heuristic method may present challenges, the benefits far outweigh the difficulties. We can unleash their full potential and prepare them for success in an ever-evolving world by providing students with opportunities to explore, discover, and construct knowledge.

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Key Tips On Problem Solving Method Of Teaching

Problem-solving skills are necessary for all strata of life, and none can be better than classroom problem-solving activities. It can be an excellent way to introduce students to problem-solving skills, get them prepped and ready to solve real problems in real-life settings.  

The ability to critically analyze a problem, map out all its elements and then prepare a solution that works is one of the most valuable skills; one must acquire in life. Educating your students about problem-solving techniques from an early age can be facilitated with in-class problem-solving activities. Such efforts encourage cognitive and social development and equip students with the tools they will need to tackle and resolve their lives.  

So, what is  a  problem-solving method of teaching ?

Problem Solving  is the act of defining a problem; determining the cause of the problem; identifying, prioritizing and selecting alternatives for a solution; and implementing a solution. In a problem-solving method, children learn by working on problems. This skill enables the students to learn new knowledge by facing the problems to be solved. It is expected of them to observe, understand, analyze, interpret, find solutions, and perform applications that lead to a holistic understanding of the concept. This method develops scientific process skills. This method helps in developing a brainstorming approach to learning concepts. 

In simple words, problem-solving is an ongoing activity in which we take what we know to discover what we do not know. It involves overcoming obstacles by generating hypotheses, testing those predictions, and arriving at satisfactory solutions. 

The problem-solving method involves three basic functions

• Seeking information
• Generating new knowledge 
• Making decisions 

This post will include key strategies to help you inculcate problem-solving skills in your students. 

First and foremostly, follow the 5-step model of problem-solving presented by Wood

Woods' problem-solving model

Identify the problem .

Allow your students to identify the system under study by interpreting the information provided in the problem statement. Then, prepare a list of what is known about the problem, and identify the knowledge needed to understand (and eventually) solve it. Once you have a list of known problems, identifying the unknown(s) becomes simpler. The unknown one is usually the answer to the problem; however, there may be other unknowns. Make sure that your students have a clear understanding of what they are expected to find. 

While teaching problem solving, it is very important to have students know how to select, interpret, and use units and symbols. Emphasize the use of units and symbols whenever appropriate. Develop a habit of using appropriate units and symbols yourself at all times. Teach your students to look for the words only and neglect or assume to help identify the constraints. 

Furthermore, help students consider from the beginning what a logical type of answer would be. What characteristics will it possess?  

Think about it

Use the next stage to ponder the identified problem. Ideally, students will develop an imaginary image of the problem at hand during this stage. They need to determine the required background knowledge from illustrations, examples and problems covered in the course and collect pertinent information such as conversion factors, constants, and tables needed to solve the problem. 

Plan a solution

Often, the type of problem will determine the type of solution. Some common problem-solving strategies are: compute; simplify; use an equation; make a model, diagram, table, or chart; or work backwards. 

Help your students choose the best strategy by reminding them again what they must find or calculate. 

Carry out the plan

Now that the major part of problem-solving has been done start executing the solution. There are possibilities that a plan may 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?  

Other tips include

Ask open-ended questions.

When a student seeks help, you might be willing to give them the answer they are looking for so you can both move on. But what is recommend is that instead of giving answers promptly, try using open-ended questions and prompts. For example: ask What do you think will happen if..? Why do you think so? What would you do if you get into such situations? Etc. 

Emphasize Process Over Product

For elementary students, reflecting on the process of solving a problem helps them develop a growth mindset. Getting an 'incorrect' response does not have to be a bad thing! What matters most is what they have done to achieve it and how they might change their approach next time. As a teacher, you can help students learn the process of reflection. 

Model The Strategies

As children learn creative problem-solving techniques, there will probably be times when they will be frustrated or uncertain. Here are just a few simple ways to model what creative problem-solving looks like and sounds like. 

• Ask questions in case you don't understand anything.
• Admit to not knowing the right answer.
• Discuss the many possible outcomes of different situations. 
• Verbalize what you feel when you come across a problem.
• Practising these strategies with your students will help create an environment where struggle, failure and growth are celebrated!

Encourage Grappling

Grappling is not confined to perseverance! This includes critical thinking, asking questions, observing evidence, asking more questions, formulating hypotheses and building a deep understanding of a problem. 

There are numerous ways to provide opportunities for students to struggle. All that includes the engineering design process is right! Examples include: 

• Engineering or creative projects
• Design-thinking challenges
• Informatics projects
• Science experiments

Make problem resolution relevant to the lives of your students

Limiting problem solving to class is a bad idea. This will affect students later in life because problem-solving is an essential part of human life, and we have had a chance to look at it from a mathematical perspective. Such problems are relevant to us, and they are not things that we are supposed to remember or learn but to put into practice in real life. These are things from which we can take very significant life lessons and apply them later in life. 

What's your strategy? How do you teach Problem-Solving to your students? Do let us know in the comments. 

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Learning to Teach Mathematics Through Problem Solving

• Open access
• Published: 21 April 2022
• Volume 57 , pages 407–423, ( 2022 )

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• Judy Bailey   ORCID: orcid.org/0000-0001-9610-9083 1  

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While there has been much research focused on beginning teachers; and mathematical problem solving in the classroom, little is known about beginning primary teachers’ learning to teach mathematics through problem solving. This longitudinal study examined what supported beginning teachers to start and sustain teaching mathematics through problem solving in their first 2 years of teaching. Findings show ‘sustaining’ required a combination of three factors: (i) participation in professional development centred on problem solving (ii) attending subject-specific complementary professional development initiatives alongside colleagues from their school; and (iii) an in-school colleague who also teaches mathematics through problem solving. If only one factor is present, in this study attending the professional development focussed on problem solving, the result was little movement towards a problem solving based pedagogy. Recommendations for supporting beginning teachers to embed problem solving are included.

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Introduction

For many years curriculum documents worldwide have positioned mathematics as a problem solving endeavour (e.g., see Australian Curriculum, Assessment and Reporting Authority, 2018 ; Ministry of Education, 2007 ). There is evidence however that even with this prolonged emphasis, problem solving has not become a significant presence in many classrooms (Felmer et al., 2019 ). Research has reported on a multitude of potential barriers, even for experienced teachers (Clarke et al., 2007 ; Holton, 2009 ). At the same time it is widely recognised that beginning teachers encounter many challenges as they start their careers, and that these challenges are particularly compelling when seeking to implement ambitious methods of teaching, such as problem solving (Wood et al., 2012 ).

Problem solving has been central to mathematics knowledge construction from the beginning of human history (Felmer et al., 2019 ). 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, the promotion and enabling of problem solving is a contentious endeavour (English & Gainsburg, 2016 ). One debate centres on whether to teach mathematics through problem solving or to teach problem solving in and of itself. Recent scholarship (and this research) leans towards teaching mathematics through problem solving as a means for students to learn mathematics and come to appreciate what it means to do mathematics (Schoenfeld, 2013 ).

Problem solving has been defined in a multitude of ways over the years. Of central importance to problem solving as it is explored in this research study is Schoenfeld’s ( 1985 ) proposition that, “if one has ready access to a solution schema for a mathematical task, that task is an exercise and not a problem” (p. 74). A more recent definition of what constitutes a mathematical problem from Mamona-Downs and Mamona ( 2013 ) also emphasises the centrality of the learner not knowing how to proceed, highlighting that problems cannot be solved by procedural effort alone. These are important distinctions because traditional school texts and programmes often position problems and problem solving as an ‘add-on’ providing a practice opportunity for a previously taught, specific procedure. Given the range of learners in any education setting an important point to also consider is that what constitutes a problem for some students may not be a problem for others (Schoenfeld, 2013 ).

A research focus exploring what supports beginning teachers’ learning about teaching mathematics through problem solving is particularly relevant at this time given calls for an increased curricular focus on mathematical practices such as problem solving (Grootenboer et al., 2021 ) and recent recommendations from an expert advisory panel on the English-medium Mathematics and Statistics curriculum in Aotearoa (Royal Society Te Apārangi, 2021 ). The ninth recommendation from this report advocates for the provision of sustained professional learning in mathematics and statistics for all teachers of Years 0–8. With regard to beginning primary teachers, the recommendation goes further suggesting that ‘mathematics and statistics professional learning’ (p. 36) be considered as compulsory in the first 2 years of teaching. This research explores what the nature of that professional learning might involve, with a focus on problem solving.

Scoping the Context for Learning and Sustaining Problem Solving

The literature reviewed for this study draws from two key fields: the nature of support and professional development effective for beginning teachers; and specialised supports helping teachers to employ problem solving pedagogies.

Beginning Teachers, Support and Professional Development

A teacher’s early years in the profession are regarded as critical in terms of constructing a professional practice (Feiman-Nemser, 2003 ) and avoiding high attrition (Karlberg & Bezzina, 2020 ). Research has established that beginning teachers need professional development opportunities geared specifically to their needs (Fantilli & McDougall, 2009 ) and their contexts (Gaikhorst, et al., 2017 ). Providing appropriate support is not an uncontentious matter with calls for institutions to come together and collaborate to provide adequate and ongoing support (Karlberg & Bezzina, 2020 ). The proposal is that support is needed from both within and beyond the beginning teacher’s school; and begins with effective pre-service teacher preparation (Keese et al., 2022 ).

Within schools where beginning teachers regard the support they receive positively, collaboration, encouragement and ‘involved colleagues’ are considered as vital; with the guidance of a 'buddy’ identified as some of the most valuable in-school support activities (Gaikhorst et al., 2014 ). Cameron et al.’s ( 2007 ) research in Aotearoa reports beginning teachers joining collaborative work cultures had greater opportunities to talk about teaching with their colleagues, share planning and resources, examine students’ work, and benefit from the collective expertise of team members.

Opportunities to participate in networks beyond the beginning teacher’s school have also been identified as being important for teacher induction (Akiri & Dori, 2021 ; Cameron et al., 2007 ). Fantilli & McDougall ( 2009 ) in their Canadian study found beginning teachers reported a need for many support and professional development opportunities including subject-specific (e.g., mathematics) workshops prior to and throughout the year. Akiri and Dori ( 2021 ) also refer to the need for specialised support from subject-specific mentors. This echoes the findings of Wood et al. ( 2012 ) who advocate that given the complexity of learning to teach mathematics, induction support specific to mathematics, and rich opportunities to learn are not only desirable but essential.

Akiri and Dori ( 2021 ) describe three levels of mentoring support for beginning teachers including individual mentoring, group mentoring and mentoring networks with all three facilitating substantive professional growth. Of relevance to this paper are individual and group mentoring. Individual mentoring involves pairing an experienced teacher with a beginning teacher, so that a beginning teacher’s learning is supported. Group mentoring involves a group of teachers working with one or more mentors, with participants receiving guidance from their mentor(s) (Akiri & Dori, 2021 ). Findings from Akiri and Dori suggest that of the varying forms of mentoring, individual mentoring contributes the most for beginning teachers’ professional learning.

Teachers Learning to Teach Mathematics Through Problem Solving

Learning to teach mathematics through problem solving begins in pre-service teacher education. It has been shown that providing pre-service teachers with opportunities to engage in problem solving as learners can be productive (Bailey, 2015 ). Opportunities to practise content-specific instructional strategies such as problem solving during student teaching has also been positively associated with first-year teachers’ enactment of problem solving (Youngs et al., 2022 ).

The move from pre-service teacher education to the classroom can be fraught for beginning teachers (Feiman-Nemser, 2003 ), and all the more so for beginning teachers attempting to teach mathematics through problem solving (Wood et al., 2012 ). In a recent study (Darragh & Radovic, 2019 ) it has been shown that an individual willingness to change to a problem-based pedagogy may not be enough to sustain a change in practice in the long term, particularly if there is a contradiction with the context and ‘norms’ (e.g., school curriculum) within which a teacher is working. Cady et al. ( 2006 ) explored the beliefs and practices of two teachers from pre-service teacher education through to becoming experienced teachers. One teacher who initially adopted reform practices reverted to traditional beliefs about the learning and teaching of mathematics. In contrast, the other teacher implemented new practices only after understanding these and gaining teaching experience. Participation in mathematically focused professional development and involvement in resource development were thought to favourably influence the second teacher.

Lesson structures have been found to support teachers learning to teach mathematics through problem solving. Sullivan et al. ( 2016 ) explored the use of a structure comprising four phases: launching, exploring, summarising and consolidating. Teachers in Australia and Aotearoa have reported the structure as productive and feasible (Ingram et al., 2019 ; Sullivan et al., 2016 ). Teaching using challenging tasks (such as in problem solving) and a structure have been shown to accommodate student diversity, a pressing concern for many teachers. Student diversity has often been managed by grouping students according to perceived levels of capability (called ability grouping). Research identifies this practice as problematic, excluding and marginalising disadvantaged groups of students (e.g., see Anthony & Hunter, 2017 ). The lesson structure explored by Sullivan et al. ( 2016 ) caters for diversity by deliberately differentiating tasks, providing enabling and extending prompts. Extending prompts are offered to students who finish an original task quickly and ideally elicit abstraction and generalisation. Enabling prompts involve reducing the number of steps, simplifying the numbers, and/or varying forms of representation for students who cannot initially proceed, with the explicit intention that students then return to the original task.

Recognising the established challenges teachers encounter when learning about teaching mathematics through problem solving, and the paucity of recent research focussing on beginning teachers learning about teaching mathematics in this way, this paper draws on data from a 2 year longitudinal study. The study was guided by the research question:

What supports beginning teachers’ implementation of a problem solving pedagogy for the teaching and learning of mathematics?

Research Methodology and Methods

Data were gathered from three beginning primary teachers who had completed a 1 year graduate diploma programme in primary teacher education the previous year. The beginning teachers had undertaken a course in mathematics education (taught by the author for half of the course) as part of the graduate diploma. An invitation to be involved in the research was sent to the graduate diploma cohort at the end of the programme. Three beginning teachers indicated their interest and remained involved for the 2 year research period. The teachers had all secured their first teaching positions, and were teaching at different year levels at three different schools. Julia (pseudonyms have been used for all names) was teaching year 0–2 (5–6 years) at a small rural school; Charlotte, year 5–6 (9–10 years) at a large urban city school; and Reine, year 7–8 (11–12 years), at another small rural school. All three beginning teachers taught at their respective schools, teaching the same year levels in both years of the study. Ethical approval was sought and given by the author’s university ethics committee. Informed consent was gained from the teachers, school principals and involved parents and children.

Participatory action research was selected as the approach in the study because of its emphasis on the participation and collaboration of all those involved (Townsend, 2013 ). Congruent with the principles of action research, activities and procedures were negotiated throughout both years in a responsive and emergent way. The author acted as a co-participant with the teachers, aiming to improve practice, to challenge and reorient thinking, and transform contexts for children’s learning (Locke et al., 2013 ). The author’s role included facilitating the research-based problem solving workshops (see below), contributing her experience as a mathematics educator and researcher. The beginning teachers were involved in making sense of their own practice related to their particular sites and context.

The first step in the research process was a focus group discussion before the beginning teachers commenced their first year of teaching. This discussion included reflecting on their learning about problem solving during the mathematics education course; and envisaging what would be helpful to support implementation. It was agreed that a series of workshops would be useful. Two were subsequently held in the first year of the study, each for three hours, at the end of terms one and two. Four workshops were held during the second year, one during each term. At the end of the first year the author suggested a small number of experienced teachers who teach mathematics through problem solving join the workshops for the second year. The presence of these teachers was envisaged to support the beginning teachers’ learning. The beginning teachers agreed, and an invitation was extended to four teachers from other schools whom the author knew (e.g., through professional subject associations). The focus of the research remained the same, namely exploring what supported beginning teachers to implement a problem solving pedagogy.

Each workshop began with sharing and oral reflections about recent problem solving experiences, including successes and challenges. Key workshop tasks included developing a shared understanding of what constitutes problem solving, participating in solving mathematical problems (modelled using a lesson structure (Sullivan et al., 2016 ), and learning techniques such as asking questions. A time for reflective writing was provided at the end of each workshop to record what had been learned and an opportunity to set goals.

During the first focus group discussion it was also decided the author would visit and observe the beginning teachers teaching a problem solving lesson (or two) in term three or four of each year. A semi-structured interview between the author and each beginning teacher took place following each observed lesson. The beginning teachers also had an opportunity to ask questions as they reflected on the lesson, and feedback was given as requested. A second focus group discussion was held at the end of the first year (an approximate midpoint in the research), and a final focus group discussion was held at the end of the second year.

All focus group discussions, problem solving workshops, observations and interviews were audio-recorded and transcribed. Field notes of workshops (recorded by the author), reflections from the beginning teachers (written at the end of each workshop), and lesson observation notes (recorded by the author) were also gathered. The final data collected included occasional emails between each beginning teacher and the author.

Data Analysis

The analysis reported in this paper drew on all data sets, primarily using inductive thematic analysis (Braun & Clarke, 2006 ). The research question guided the key question for analysis, namely: What supports beginning teachers’ implementation of a problem solving pedagogy for the teaching and learning of mathematics? Alongside this question, consideration was also given to the challenges beginning teachers encountered as they implemented a problem solving pedagogy. Data familiarisation was developed through reading and re-reading the whole body of data. This process informed data analysis and the content for each subsequent workshop and focus group discussions. Colour-coding and naming of themes included comparing and contrasting data from each beginning teacher and throughout the 2-year period. As a theme was constructed (Braun & Clarke, 2006 ) subsequent data was checked to ascertain whether the theme remained valid and/or whether it changed during the 2 years. Three key themes emerged revealing what supported the beginning teachers’ developing problem solving pedagogy, and these constitute the focus for this paper.

Mindful of the time pressures beginning teachers experience in their early years, the author undertook responsibility for data analysis. The author’s understanding of the unfolding ‘story’ of each beginning teacher’s experiences and the emerging themes were shared with the beginning teachers, usually at the beginning of a workshop, focus group discussion or observation. Through this process the author’s understandings were checked and clarified. This iterative process of member checking (Lincoln & Guba, 1985 ) began at a mid-point during the first year, once a significant body of data had been gathered. At a later point in the analysis and writing, the beginning teachers also had an opportunity to read, check and/or amend quotes chosen to exemplify their thinking and experiences.

Findings and Discussion

In this section the three beginning teachers’ experiences at the start of the 2 year research timeframe is briefly described, followed by the first theme centred on the use of a lesson structure including prompts for differentiation. The second and third themes are presented together, starting with a brief outline of each beginning teacher’s ‘story’ providing the context within which the themes emerged. Sharing the ‘story’ of each beginning teacher and including their ‘voice’ through quotes acknowledges them and their experiences as central to this research.

The beginning teachers’ pre-service teacher education set the scene for learning about teaching mathematics through problem solving. A detailed list brainstormed during the first focus group discussion suggested a developing understanding from their shared pre-service mathematics education course. In their first few weeks of teaching, all three beginning teachers implemented a few problems. It transpired however this inclusion of problem solving occurred only while children were being assessed and grouped. Following this, all three followed a traditional format of skill-based (with a focus on number) mathematics, taught using ability groups. The beginning teachers’ trajectories then varied with Julia and Reine both eventually adopting a pedagogy primarily based on problem solving, while Charlotte employed a traditional skill-based mathematics using a combination of whole class and small group teaching.

A Lesson Structure that Caters for Diversity Supports Early Efforts

Data show that developing familiarity with a lesson structure including prompts for differentiation supported the beginning teachers’ early efforts with a problem solving pedagogy. This addressed a key issue that emerged during the first workshop. During the workshop while a ‘list’ of ideas for teaching a problem solving lesson was co-constructed, considerable concern was expressed about catering for a range of learners when introducing and working with a problem. For example, Charlotte queried, “ Well, what happens when you are trying to do something more complicated, and we’re (referring to children) sitting here going, ‘I’ve no idea what you're talking about” ? Reine suggested keeping some children with the teacher, thinking he would say, “ If you’re unsure of any part stay behind” . He was unsure however about how he would then maintain the integrity of the problem.

It was in light of this discussion that a lesson structure with differentiated prompts (Sullivan et al., 2016 ) was introduced, experienced and reflected on during the second workshop. While the co-constructed list developed during the first workshop had included many components of Sullivan’s lesson structure, (e.g., a consideration of ‘extensions’) there had been no mention of ‘enabling prompts’. Now, with the inclusion of both enabling and extending prompts, the beginning teachers’ discussion revealed them starting to more fully envisage the possibilities of using a problem solving approach, and being able to cater for all children. Reine commented that, “… you can give the entire class a problem, you've just got to have a plan, [and] your enabling and extension prompts” . Charlotte was also now considering and valuing the possibility of having a whole class work on the same problem. She said, “I think … it’s important and it’s useful for your whole class to be working on the same thing. And … have enablers and extenders to make sure that everyone feels successful” . Julia also referred to the planning prompts. She thought it would be key to “plan it well so that we’ve got enabling and extending prompts” .

Successful Problem Solving Lessons

Following the second workshop all three beginning teachers were observed teaching a lesson using the structure. These lessons delighted the beginning teachers, with them noting prolonged engagement of children, the children’s learning and being able to cater for all learners. Reine commented on how excited and engaged the children were, saying they were, “ just so enthusiastic about it ”. In Charlotte’s words, “ it really worked ”, and Julia enthusiastically pondered this could be “ the only way you teach maths !”.

During the focus group discussion at the end of the first year, all three reflected on the value of the lesson structure. Reine called it a ‘framework’ commenting,

I like the framework. So from start to finish, how you go through that whole lesson. So how you set it up and then you go through the phases… I like the prompts that we went through…. knowing where you could go, if they’re like, ‘What do I do?’ And then if they get it too easy then ‘Where can you go?’ So you've got all these little avenues.

Charlotte also valued the lesson structure for the breadth of learning that could occur, explaining,

… it really helped, and really worked. So I found that useful for me and my class ‘cause they really understood. And I think also making sure that you know all the ins and outs of a problem. So where could they go? What do you need to know? What do they need to know?

While the beginning teachers’ pre-service teacher education and the subsequent research process, including the use of the lesson structure, supported the beginning teachers’ early efforts teaching mathematics through problem solving, two key factors further enabled two of the beginning teachers (Julia and Reine) to sustain a problem solving pedagogy. These were:

Being involved in complementary mathematics professional development alongside members of their respective school staff (a form of group mentoring); and

Having a colleague in the same school teaching mathematics through problem solving (a form of individual mentoring).

Charlotte did not have these opportunities and she indicated this limited her implementation. Data for these findings for each teacher are presented below.

Complementary Professional Development and Problem Solving Colleague in Same School

Julia began to significantly implement problem solving from the second term in the first year. This coincided with her attending a 2-day workshop (with staff from her school) that focused on the use of problem solving to support children who are not achieving at expected levels (see ALiM: Accelerated Learning in Maths—Ministry of Education, 2022 ). She explained, “ … I did the PD with (colleague’s name), which was really helpful. And we did lots of talking about rich learning tasks and problem solving tasks…. And what it means ”. Following this, Julia reported using rich tasks and problem solving in her mathematics teaching in a regular (at least weekly) and ongoing way.

During the observation in term three of the first year Julia again referred to the impact of having a colleague also teaching mathematics through problem solving. When asked what she believed had supported her to become a teacher who teaches mathematics in this way she firstly identified her involvement in the research project, and then spoke about her colleague. She said, “ I’m really lucky one of our other teachers is doing the ALiM project… So we’re kind of bouncing off each other a little bit with resources and activities, and things like that. So that’s been really good ”.

At the beginning of the second year, Julia reiterated this point again. On this occasion she said having a colleague teaching mathematics through problem solving, “ made a huge difference for me last year ”, explaining the value included having someone to talk with on a daily basis. Mid-way through the second year Julia repeated her opinion about the value of frequent contact with a practising problem solving colleague. Whereas her initial comments spoke of the impact in terms of being “ a little bit ”, later references recount these as ‘ huge ’ and ‘ enabling ’. She described:

a huge effect… it enabled me. Cause I mean these workshops are really helpful. But when it’s only once a term, having [colleague] there just enabled me to kind of bounce ideas off. And if I did a lesson that didn’t work very well, we could talk about why that was, and actually talk about what the learning was instead…. . It was being able to reflect together, but also share ideas. It was amazing.

Julia’s comments raise two points. It is likely that participating in the ALiM professional development (which could be conceived as a form of group mentoring) consolidated the learning she first encountered during pre-service teacher education and later extended through her involvement in the research. Having a colleague (in essence, an individual mentor) within the same school teaching mathematics through problem solving appears to be another factor that supported Julia to implement problem solving in a more sustained way. Julia’s comments allude to a number of reasons for this, including: (i) the more frequent discussion opportunities with a colleague who understands what it means for children to learn mathematics through problem solving; (ii) being able to share and plan suitable activities and resources; and (iii) as a means for reflection, particularly when challenges were encountered.

Reine’s mathematics programme throughout the first year was based on ability groups and could be described as traditional. He occasionally used some mathematical problems as ‘extension activities’ for ‘higher level’ children, or as ‘fillers’. In the second year, Reine moved to working with mixed ability groups (where students work together in small groups with varying levels of perceived capability) and initially implemented problem solving approximately once a fortnight. In thinking back to these lessons he commented, “ We weren’t really unpacking one problem properly, it was just lots of busy stuff ”. A significant shift occurred in Reine’s practice to teaching mathematics primarily by problem solving towards the last half of the second year. He explained, “ I really ramped up towards terms three and four, where it’s more picking one problem across the whole maths class but being really, really conscious of that problem. Low entry, high ceiling, and doing more of it too ”.

Reine attributed this change to a number of factors. In response to a question about what he considered led to the change he explained,

… having this, talking about this stuff, trialling it and then with our PD at school with the research into ability grouping... We’ve got a lot of PD saying why it can be harmful to group on ability, and that’s been that last little kick I needed, I think. And with other teachers trialling this as well. Our senior teacher has flipped her whole maths program and just does problem solving.

Like Julia, Reine firstly referred to his involvement in the research project including having opportunities to try problems in his class and discuss his experiences within the research group. He then told of a colleague teaching at his school leading school-wide professional development focussed on the pitfalls of ability grouping in mathematics (e.g., see Clarke, 2021 ) and instead using problem solving tasks. He also referred to having another teacher also teaching mathematics through problem solving. It is interesting to consider that having positive experiences in pre-service teacher education, the positive and encouraging support of colleagues (Reine’s principal and co-teacher in both years), regular participation in ongoing professional development (the problem solving workshops), and having a highly successful one-off problem solving teaching experience (the first year observation) were not enough for Reine to meaningfully sustain problem solving in his first year of teaching.

As for Julia, pivotal factors leading to a sustaining of problem solving teaching practice in the second year included complementary mathematics professional development (a form of group mentoring) and at least one other teacher (acting as an individual mentor) in the same school teaching mathematics through problem solving. It could be argued that pre-service teacher education and the problem solving workshops ‘paved the way’ for Julia and Reine to make a change. However, for both, the complementary professional development and presence of a colleague also teaching through problem solving were pivotal. It is also interesting to note that three of the four experienced teachers in the larger research group taught at the same level as Reine (see Table 1 below) yet he did not relate this to the significant change in his practice observed towards the end of the second year.

Charlotte’s mathematics programme during the first year was also traditional, teaching skill-based mathematics using ability groups. At the beginning of the second year Charlotte moved to teaching her class as a whole group, using flexible grouping as needed (children are grouped together in response to learning needs with regard to a specific idea at a point in time, rather than perceived notions of ability). She reported that she occasionally taught a lesson using problem solving in the first year, and approximately once or twice a term in the second year. Charlotte did not have opportunities for professional development in mathematics nor did she have a colleague in the same school teaching mathematics through problem solving. Pondering this, Charlotte said,

It would have been helpful if I had someone else in my school doing the same thing. I just thought about when you were saying the other lady was doing it [referring to Julia’s colleague]. You know, someone that you can just kind of back-and-forth like. I find with Science, I usually plan with this other lady, and we share ideas and plan together. We come up with some really cool stuff whereas I don’t really have the same thing for this.

Based on her experiences with teaching science it is clear Charlotte recognised the value of working alongside a colleague. In this, her view aligns with what Julia and Reine experienced.

Table 1 provides a summary of the variables for each beginning teacher, and whether a sustained implementation of teaching mathematics through problem solving occurred.

The table shows two variables common to Julia and Reine, the beginning teachers who began and sustained problem solving. They both participated in complementary professional development with colleagues from their school, and the presence of a colleague, also at their school, teaching mathematics through problem solving. Given that Julia was able to implement problem solving in the absence of a ‘research workshop colleague’ teaching at the same year level, and Reine’s lack of comment about the potential impact of this, suggests that this was not a key factor enabling a sustained implementation of problem solving.

Attributing the changes in Julia and Reine’s teaching practice primarily to their involvement in complementary professional development attended by members of their school staff, and the presence of at least one other teacher teaching mathematics through problem solving in their school, is further supported by a consideration of the timing of the changes. The data shows that while Julia could be considered an ‘early adopter’, Reine changed his practice reasonably late in the 2 year period. Julia’s early adoption of teaching mathematics through problem solving coincided with her involvement, early in the 2 years, in the professional development and opportunity to work alongside a problem solving practising colleague. Reine encountered these similar conditions towards the end of the 2 years and it is notable that this was the point at which he changed his practice. That problem solving did not become embedded or frequent within Charlotte’s mathematics programme tends to support the argument.

Understanding what supports primary teachers to teach mathematics through problem solving at the beginning of their careers is important because all students, including those taught by beginning teachers, need opportunities to develop high-level thinking, reasoning, and problem solving skills. It is also important in light of recent calls for mathematics curricula to include more emphasis on mathematical practices (such as problem solving) (e.g., see Grootenboer et al., 2021 ); and the Royal Society Te Apārangi report ( 2021 ). Findings from this research suggest that learning about problem solving during pre-service teacher education is enough for beginning teachers to trial teaching mathematics in this way. Early efforts were supported by gaining experience with a lesson structure that specifically attends to diversity. The lesson structure prompted the beginning teachers to anticipate different children’s responses, and consider how they would respond to these. An increased confidence and sense of security to trial teaching mathematics through problem solving was enabled, based on their more in-depth preparation. Beginning teachers finding the lesson structure useful extends the findings of Sullivan et al. ( 2016 ) in Australia and Ingram et al. ( 2019 ) in Aotearoa to include less experienced teachers.

In order for teaching mathematics through problem solving to be sustained however, a combination of three factors, subsequent to pre-service teacher education, was needed: (i) active participation in problem solving workshops (in this context provided by the research-based problem solving workshops); (ii) attending complementary professional development initiatives alongside colleagues from their school (a form of group mentoring); and (iii) the presence of an in-school colleague who also teaches mathematics through problem solving (a form of individual mentoring). It seems possible these three factors acted synergistically resulting in Julia and Reine being able to sustain implementation. If only one factor is present, in this study attending the problem solving workshops, and despite a genuine interest in using a problem based pedagogy, the result was limited movement towards this way of teaching.

Akiri and Dori ( 2021 ) have reported that individual mentoring contributes the most to beginning teachers’ professional growth. In a manner consistent with these findings, an in-school colleague (who in essence was acting as an individual mentor) played a critical role in supporting Reine and Julia. However, while Akiri and Dori, amongst others (e.g., Cameron et al., 2007 ; Karlberg & Bezzina, 2020 ), have identified the value of supportive, approachable colleagues, for both Julia and Reine it was important that their colleague was supportive and approachable, and actively engaged in teaching mathematics through problem solving. Having supportive and approachable colleagues, as Reine experienced in his first year, on their own were not enough to support a sustained problem solving pedagogy.

Implications for Productive Professional Learning and Development

This study sought to explore the conditions that supported problem solving for beginning teachers, each in their unique context and from their perspective. The research did not examine how the teaching of mathematics through problem solving affected children’s learning. However, multiple sets of data were collected and analysed over a 2-year period. While it is neither possible nor appropriate to make claims as to generalisability some suggestions for productive beginning teacher professional learning and development are offered.

Given the first years of teaching constitute a particular and critical phase of teacher learning (Karlberg & Bezzina, 2020 ) and the findings from this research, it is imperative that well-funded, subject-focussed support occurs throughout a beginning teacher’s first 2 years of teaching. This is consistent with the ninth recommendation in the Royal Society Te Apārangi report ( 2021 ) suggesting compulsory professional learning during the induction period (2 years in Aotearoa New Zealand). Participation in subject-specific professional development has been recognised to favourably influence new teachers’ efforts to adopt reform practices such as problem solving (Cady et al., 2006 ).

Findings from this study suggest professional development opportunities that complement each other support beginning teacher learning. In the first instance complementarity needs to be with what beginning teachers have learned during their pre-service teacher education. In this study, the research-based problem solving workshops served this role. Complementarity between varying forms of professional development also appears to be important. Furthermore, as indicated by Julia and Reine’s experiences, subsequent professional development need not be on exactly the same topic. Rather, it can be complementary in the sense that there is an underlying congruence in philosophy and/or focus on a particular issue. For example, it emerged in the problem solving workshops, that being able to cater for diversity was a central concern for the beginning teachers. Attending to this issue within the problem solving workshops via the introduction of a lesson structure that enabled differentiation, was congruent with the nature of the professional development in the two schools: ALiM in Julia’s school, and mixed ability grouping and teaching mathematics through problem solving in Reine’s school. All three of these settings were focussed on positively responding to diversity in learning needs.

The presence of a colleague within the same school teaching mathematics through problem solving also appears to be pivotal. This is consistent with Darragh and Radovic ( 2019 ) who have shown the significant impact a teacher’s school context has on their potential to sustain problem based pedagogies in mathematics. Given that problem solving is not prevalent in many primary classrooms, it would seem clear that colleagues who have yet to learn about teaching mathematics through problem solving, particularly those that have a role supporting beginning teachers, will also require access to professional development opportunities. It seems possible that beginning and experienced teachers learning together is a potential pathway forward. Finding such pathways will be critical if mathematical problem solving is to be consistently implemented in primary classrooms.

Finally, these implications together with calls for institutions to collaborate to provide adequate and ongoing support for new teachers (Karlberg & Bezzina, 2020 ) suggest there is a need for pre-service teacher educators, professional development providers and the Teaching Council of Aotearoa New Zealand to work together to support beginning teachers’ starting and sustaining teaching mathematics through problem solving pedagogies.

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Bailey, J. Learning to Teach Mathematics Through Problem Solving. NZ J Educ Stud 57 , 407–423 (2022). https://doi.org/10.1007/s40841-022-00249-0

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Art and Science of Teaching / Problem Solving in Seven Steps

Step 1: Determine whether you have a problem and whether it's worth solving

Step 2: affirm positive beliefs regarding your ability to solve the problem, step 3: clarify the obstacle and identify possible solutions, step 4: determine each solution's likelihood of success and consider the resources required, step 5: try out the solution that has the greatest chance of success, step 6: if your solution doesn't work, try a different one, step 7: if you can't find a solution, identify an alternative goal, make it explicit.

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