a descriptive approach to problem solving

How to improve your problem solving skills and build effective problem solving strategies

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Problem solving strategies

What skills do i need to be an effective problem solver, how can i improve my problem solving skills.

Problem solving strategies are methods of approaching and facilitating the process of problem-solving with a set of techniques , actions, and processes. Different strategies are more effective if you are trying to solve broad problems such as achieving higher growth versus more focused problems like, how do we improve our customer onboarding process?

Broadly, the problem solving steps outlined above should be included in any problem solving strategy though choosing where to focus your time and what approaches should be taken is where they begin to differ. You might find that some strategies ask for the problem identification to be done prior to the session or that everything happens in the course of a one day workshop.

The key similarity is that all good problem solving strategies are structured and designed. Four hours of open discussion is never going to be as productive as a four-hour workshop designed to lead a group through a problem solving process.

Good problem solving strategies are tailored to the team, organization and problem you will be attempting to solve. Here are some example problem solving strategies you can learn from or use to get started.

Use a workshop to lead a team through a group process

Often, the first step to solving problems or organizational challenges is bringing a group together effectively. Most teams have the tools, knowledge, and expertise necessary to solve their challenges – they just need some guidance in how to use leverage those skills and a structure and format that allows people to focus their energies.

Facilitated workshops are one of the most effective ways of solving problems of any scale. By designing and planning your workshop carefully, you can tailor the approach and scope to best fit the needs of your team and organization.

Problem solving workshop

Creating a bespoke, tailored process
Tackling problems of any size
Building in-house workshop ability and encouraging their use

Workshops are an effective strategy for solving problems. By using tried and test facilitation techniques and methods, you can design and deliver a workshop that is perfectly suited to the unique variables of your organization. You may only have the capacity for a half-day workshop and so need a problem solving process to match.

By using our session planner tool and importing methods from our library of 700+ facilitation techniques, you can create the right problem solving workshop for your team. It might be that you want to encourage creative thinking or look at things from a new angle to unblock your groups approach to problem solving. By tailoring your workshop design to the purpose, you can help ensure great results.

One of the main benefits of a workshop is the structured approach to problem solving. Not only does this mean that the workshop itself will be successful, but many of the methods and techniques will help your team improve their working processes outside of the workshop.

We believe that workshops are one of the best tools you can use to improve the way your team works together. Start with a problem solving workshop and then see what team building, culture or design workshops can do for your organization!

Run a design sprint

Great for:

aligning large, multi-discipline teams
quickly designing and testing solutions
tackling large, complex organizational challenges and breaking them down into smaller tasks

By using design thinking principles and methods, a design sprint is a great way of identifying, prioritizing and prototyping solutions to long term challenges that can help solve major organizational problems with quick action and measurable results.

Some familiarity with design thinking is useful, though not integral, and this strategy can really help a team align if there is some discussion around which problems should be approached first.

The stage-based structure of the design sprint is also very useful for teams new to design thinking. The inspiration phase, where you look to competitors that have solved your problem, and the rapid prototyping and testing phases are great for introducing new concepts that will benefit a team in all their future work.

It can be common for teams to look inward for solutions and so looking to the market for solutions you can iterate on can be very productive. Instilling an agile prototyping and testing mindset can also be great when helping teams move forwards – generating and testing solutions quickly can help save time in the long run and is also pretty exciting!

Break problems down into smaller issues

Organizational challenges and problems are often complicated and large scale in nature. Sometimes, trying to resolve such an issue in one swoop is simply unachievable or overwhelming. Try breaking down such problems into smaller issues that you can work on step by step. You may not be able to solve the problem of churning customers off the bat, but you can work with your team to identify smaller effort but high impact elements and work on those first.

This problem solving strategy can help a team generate momentum, prioritize and get some easy wins. It’s also a great strategy to employ with teams who are just beginning to learn how to approach the problem solving process. If you want some insight into a way to employ this strategy, we recommend looking at our design sprint template below!

Use guiding frameworks or try new methodologies

Some problems are best solved by introducing a major shift in perspective or by using new methodologies that encourage your team to think differently.

Props and tools such as Methodkit , which uses a card-based toolkit for facilitation, or Lego Serious Play can be great ways to engage your team and find an inclusive, democratic problem solving strategy. Remember that play and creativity are great tools for achieving change and whatever the challenge, engaging your participants can be very effective where other strategies may have failed.

LEGO Serious Play

Improving core problem solving skills
Thinking outside of the box
Encouraging creative solutions

LEGO Serious Play is a problem solving methodology designed to get participants thinking differently by using 3D models and kinesthetic learning styles. By physically building LEGO models based on questions and exercises, participants are encouraged to think outside of the box and create their own responses.

Collaborate LEGO Serious Play exercises are also used to encourage communication and build problem solving skills in a group. By using this problem solving process, you can often help different kinds of learners and personality types contribute and unblock organizational problems with creative thinking.

Problem solving strategies like LEGO Serious Play are super effective at helping a team solve more skills-based problems such as communication between teams or a lack of creative thinking. Some problems are not suited to LEGO Serious Play and require a different problem solving strategy.

Card Decks and Method Kits

New facilitators or non-facilitators
Approaching difficult subjects with a simple, creative framework
Engaging those with varied learning styles

Card decks and method kids are great tools for those new to facilitation or for whom facilitation is not the primary role. Card decks such as the emotional culture deck can be used for complete workshops and in many cases, can be used right out of the box. Methodkit has a variety of kits designed for scenarios ranging from personal development through to personas and global challenges so you can find the right deck for your particular needs.

Having an easy to use framework that encourages creativity or a new approach can take some of the friction or planning difficulties out of the workshop process and energize a team in any setting. Simplicity is the key with these methods. By ensuring everyone on your team can get involved and engage with the process as quickly as possible can really contribute to the success of your problem solving strategy.

Source external advice

Looking to peers, experts and external facilitators can be a great way of approaching the problem solving process. Your team may not have the necessary expertise, insights of experience to tackle some issues, or you might simply benefit from a fresh perspective. Some problems may require bringing together an entire team, and coaching managers or team members individually might be the right approach. Remember that not all problems are best resolved in the same manner.

If you’re a solo entrepreneur, peer groups, coaches and mentors can also be invaluable at not only solving specific business problems, but in providing a support network for resolving future challenges. One great approach is to join a Mastermind Group and link up with like-minded individuals and all grow together. Remember that however you approach the sourcing of external advice, do so thoughtfully, respectfully and honestly. Reciprocate where you can and prepare to be surprised by just how kind and helpful your peers can be!

Mastermind Group

Solo entrepreneurs or small teams with low capacity
Peer learning and gaining outside expertise
Getting multiple external points of view quickly

Problem solving in large organizations with lots of skilled team members is one thing, but how about if you work for yourself or in a very small team without the capacity to get the most from a design sprint or LEGO Serious Play session?

A mastermind group – sometimes known as a peer advisory board – is where a group of people come together to support one another in their own goals, challenges, and businesses. Each participant comes to the group with their own purpose and the other members of the group will help them create solutions, brainstorm ideas, and support one another.

Mastermind groups are very effective in creating an energized, supportive atmosphere that can deliver meaningful results. Learning from peers from outside of your organization or industry can really help unlock new ways of thinking and drive growth. Access to the experience and skills of your peers can be invaluable in helping fill the gaps in your own ability, particularly in young companies.

A mastermind group is a great solution for solo entrepreneurs, small teams, or for organizations that feel that external expertise or fresh perspectives will be beneficial for them. It is worth noting that Mastermind groups are often only as good as the participants and what they can bring to the group. Participants need to be committed, engaged and understand how to work in this context.

Coaching and mentoring

Focused learning and development
Filling skills gaps
Working on a range of challenges over time

Receiving advice from a business coach or building a mentor/mentee relationship can be an effective way of resolving certain challenges. The one-to-one format of most coaching and mentor relationships can really help solve the challenges those individuals are having and benefit the organization as a result.

A great mentor can be invaluable when it comes to spotting potential problems before they arise and coming to understand a mentee very well has a host of other business benefits. You might run an internal mentorship program to help develop your team’s problem solving skills and strategies or as part of a large learning and development program. External coaches can also be an important part of your problem solving strategy, filling skills gaps for your management team or helping with specific business issues.

Now we’ve explored the problem solving process and the steps you will want to go through in order to have an effective session, let’s look at the skills you and your team need to be more effective problem solvers.

Problem solving skills are highly sought after, whatever industry or team you work in. Organizations are keen to employ people who are able to approach problems thoughtfully and find strong, realistic solutions. Whether you are a facilitator , a team leader or a developer, being an effective problem solver is a skill you’ll want to develop.

Problem solving skills form a whole suite of techniques and approaches that an individual uses to not only identify problems but to discuss them productively before then developing appropriate solutions.

Here are some of the most important problem solving skills everyone from executives to junior staff members should learn. We’ve also included an activity or exercise from the SessionLab library that can help you and your team develop that skill.

If you’re running a workshop or training session to try and improve problem solving skills in your team, try using these methods to supercharge your process!

Active listening

Active listening is one of the most important skills anyone who works with people can possess. In short, active listening is a technique used to not only better understand what is being said by an individual, but also to be more aware of the underlying message the speaker is trying to convey. When it comes to problem solving, active listening is integral for understanding the position of every participant and to clarify the challenges, ideas and solutions they bring to the table.

Some active listening skills include:

Paying complete attention to the speaker.
Removing distractions.
Avoid interruption.
Taking the time to fully understand before preparing a rebuttal.
Responding respectfully and appropriately.
Demonstrate attentiveness and positivity with an open posture, making eye contact with the speaker, smiling and nodding if appropriate. Show that you are listening and encourage them to continue.
Be aware of and respectful of feelings. Judge the situation and respond appropriately. You can disagree without being disrespectful.
Observe body language.
Paraphrase what was said in your own words, either mentally or verbally.
Remain neutral.
Reflect and take a moment before responding.
Ask deeper questions based on what is said and clarify points where necessary.

Active Listening #hyperisland #skills #active listening #remote-friendly This activity supports participants to reflect on a question and generate their own solutions using simple principles of active listening and peer coaching. It’s an excellent introduction to active listening but can also be used with groups that are already familiar with it. Participants work in groups of three and take turns being: “the subject”, the listener, and the observer.

Analytical skills

All problem solving models require strong analytical skills, particularly during the beginning of the process and when it comes to analyzing how solutions have performed.

Analytical skills are primarily focused on performing an effective analysis by collecting, studying and parsing data related to a problem or opportunity.

It often involves spotting patterns, being able to see things from different perspectives and using observable facts and data to make suggestions or produce insight.

Analytical skills are also important at every stage of the problem solving process and by having these skills, you can ensure that any ideas or solutions you create or backed up analytically and have been sufficiently thought out.

Nine Whys #innovation #issue analysis #liberating structures With breathtaking simplicity, you can rapidly clarify for individuals and a group what is essentially important in their work. You can quickly reveal when a compelling purpose is missing in a gathering and avoid moving forward without clarity. When a group discovers an unambiguous shared purpose, more freedom and more responsibility are unleashed. You have laid the foundation for spreading and scaling innovations with fidelity.

Collaboration

Trying to solve problems on your own is difficult. Being able to collaborate effectively, with a free exchange of ideas, to delegate and be a productive member of a team is hugely important to all problem solving strategies.

Remember that whatever your role, collaboration is integral, and in a problem solving process, you are all working together to find the best solution for everyone.

Marshmallow challenge with debriefing #teamwork #team #leadership #collaboration In eighteen minutes, teams must build the tallest free-standing structure out of 20 sticks of spaghetti, one yard of tape, one yard of string, and one marshmallow. The marshmallow needs to be on top. The Marshmallow Challenge was developed by Tom Wujec, who has done the activity with hundreds of groups around the world. Visit the Marshmallow Challenge website for more information. This version has an extra debriefing question added with sample questions focusing on roles within the team.

Communication

Being an effective communicator means being empathetic, clear and succinct, asking the right questions, and demonstrating active listening skills throughout any discussion or meeting.

In a problem solving setting, you need to communicate well in order to progress through each stage of the process effectively. As a team leader, it may also fall to you to facilitate communication between parties who may not see eye to eye. Effective communication also means helping others to express themselves and be heard in a group.

Bus Trip #feedback #communication #appreciation #closing #thiagi #team This is one of my favourite feedback games. I use Bus Trip at the end of a training session or a meeting, and I use it all the time. The game creates a massive amount of energy with lots of smiles, laughs, and sometimes even a teardrop or two.

Creative problem solving skills can be some of the best tools in your arsenal. Thinking creatively, being able to generate lots of ideas and come up with out of the box solutions is useful at every step of the process.

The kinds of problems you will likely discuss in a problem solving workshop are often difficult to solve, and by approaching things in a fresh, creative manner, you can often create more innovative solutions.

Having practical creative skills is also a boon when it comes to problem solving. If you can help create quality design sketches and prototypes in record time, it can help bring a team to alignment more quickly or provide a base for further iteration.

The paper clip method #sharing #creativity #warm up #idea generation #brainstorming The power of brainstorming. A training for project leaders, creativity training, and to catalyse getting new solutions.

Critical thinking

Critical thinking is one of the fundamental problem solving skills you’ll want to develop when working on developing solutions. Critical thinking is the ability to analyze, rationalize and evaluate while being aware of personal bias, outlying factors and remaining open-minded.

Defining and analyzing problems without deploying critical thinking skills can mean you and your team go down the wrong path. Developing solutions to complex issues requires critical thinking too – ensuring your team considers all possibilities and rationally evaluating them.

Agreement-Certainty Matrix #issue analysis #liberating structures #problem solving You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic . A problem is simple when it can be solved reliably with practices that are easy to duplicate. It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably. A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail. Chaotic is when the context is too turbulent to identify a path forward. A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.” The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Data analysis

Though it shares lots of space with general analytical skills, data analysis skills are something you want to cultivate in their own right in order to be an effective problem solver.

Being good at data analysis doesn’t just mean being able to find insights from data, but also selecting the appropriate data for a given issue, interpreting it effectively and knowing how to model and present that data. Depending on the problem at hand, it might also include a working knowledge of specific data analysis tools and procedures.

Having a solid grasp of data analysis techniques is useful if you’re leading a problem solving workshop but if you’re not an expert, don’t worry. Bring people into the group who has this skill set and help your team be more effective as a result.

Decision making

All problems need a solution and all solutions require that someone make the decision to implement them. Without strong decision making skills, teams can become bogged down in discussion and less effective as a result.

Making decisions is a key part of the problem solving process. It’s important to remember that decision making is not restricted to the leadership team. Every staff member makes decisions every day and developing these skills ensures that your team is able to solve problems at any scale. Remember that making decisions does not mean leaping to the first solution but weighing up the options and coming to an informed, well thought out solution to any given problem that works for the whole team.

Lightning Decision Jam (LDJ) #action #decision making #problem solving #issue analysis #innovation #design #remote-friendly The problem with anything that requires creative thinking is that it’s easy to get lost—lose focus and fall into the trap of having useless, open-ended, unstructured discussions. Here’s the most effective solution I’ve found: Replace all open, unstructured discussion with a clear process. What to use this exercise for: Anything which requires a group of people to make decisions, solve problems or discuss challenges. It’s always good to frame an LDJ session with a broad topic, here are some examples: The conversion flow of our checkout Our internal design process How we organise events Keeping up with our competition Improving sales flow

Dependability

Most complex organizational problems require multiple people to be involved in delivering the solution. Ensuring that the team and organization can depend on you to take the necessary actions and communicate where necessary is key to ensuring problems are solved effectively.

Being dependable also means working to deadlines and to brief. It is often a matter of creating trust in a team so that everyone can depend on one another to complete the agreed actions in the agreed time frame so that the team can move forward together. Being undependable can create problems of friction and can limit the effectiveness of your solutions so be sure to bear this in mind throughout a project.

Team Purpose & Culture #team #hyperisland #culture #remote-friendly This is an essential process designed to help teams define their purpose (why they exist) and their culture (how they work together to achieve that purpose). Defining these two things will help any team to be more focused and aligned. With support of tangible examples from other companies, the team members work as individuals and a group to codify the way they work together. The goal is a visual manifestation of both the purpose and culture that can be put up in the team’s work space.

Emotional intelligence

Emotional intelligence is an important skill for any successful team member, whether communicating internally or with clients or users. In the problem solving process, emotional intelligence means being attuned to how people are feeling and thinking, communicating effectively and being self-aware of what you bring to a room.

There are often differences of opinion when working through problem solving processes, and it can be easy to let things become impassioned or combative. Developing your emotional intelligence means being empathetic to your colleagues and managing your own emotions throughout the problem and solution process. Be kind, be thoughtful and put your points across care and attention.

Being emotionally intelligent is a skill for life and by deploying it at work, you can not only work efficiently but empathetically. Check out the emotional culture workshop template for more!

Facilitation

As we’ve clarified in our facilitation skills post, facilitation is the art of leading people through processes towards agreed-upon objectives in a manner that encourages participation, ownership, and creativity by all those involved. While facilitation is a set of interrelated skills in itself, the broad definition of facilitation can be invaluable when it comes to problem solving. Leading a team through a problem solving process is made more effective if you improve and utilize facilitation skills – whether you’re a manager, team leader or external stakeholder.

The Six Thinking Hats #creative thinking #meeting facilitation #problem solving #issue resolution #idea generation #conflict resolution The Six Thinking Hats are used by individuals and groups to separate out conflicting styles of thinking. They enable and encourage a group of people to think constructively together in exploring and implementing change, rather than using argument to fight over who is right and who is wrong.

Flexibility

Being flexible is a vital skill when it comes to problem solving. This does not mean immediately bowing to pressure or changing your opinion quickly: instead, being flexible is all about seeing things from new perspectives, receiving new information and factoring it into your thought process.

Flexibility is also important when it comes to rolling out solutions. It might be that other organizational projects have greater priority or require the same resources as your chosen solution. Being flexible means understanding needs and challenges across the team and being open to shifting or arranging your own schedule as necessary. Again, this does not mean immediately making way for other projects. It’s about articulating your own needs, understanding the needs of others and being able to come to a meaningful compromise.

The Creativity Dice #creativity #problem solving #thiagi #issue analysis Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

Working in any group can lead to unconscious elements of groupthink or situations in which you may not wish to be entirely honest. Disagreeing with the opinions of the executive team or wishing to save the feelings of a coworker can be tricky to navigate, but being honest is absolutely vital when to comes to developing effective solutions and ensuring your voice is heard.

Remember that being honest does not mean being brutally candid. You can deliver your honest feedback and opinions thoughtfully and without creating friction by using other skills such as emotional intelligence.

Explore your Values #hyperisland #skills #values #remote-friendly Your Values is an exercise for participants to explore what their most important values are. It’s done in an intuitive and rapid way to encourage participants to follow their intuitive feeling rather than over-thinking and finding the “correct” values. It is a good exercise to use to initiate reflection and dialogue around personal values.

Initiative

The problem solving process is multi-faceted and requires different approaches at certain points of the process. Taking initiative to bring problems to the attention of the team, collect data or lead the solution creating process is always valuable. You might even roadtest your own small scale solutions or brainstorm before a session. Taking initiative is particularly effective if you have good deal of knowledge in that area or have ownership of a particular project and want to get things kickstarted.

That said, be sure to remember to honor the process and work in service of the team. If you are asked to own one part of the problem solving process and you don’t complete that task because your initiative leads you to work on something else, that’s not an effective method of solving business challenges.

15% Solutions #action #liberating structures #remote-friendly You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference. 15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change. With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

Impartiality

A particularly useful problem solving skill for product owners or managers is the ability to remain impartial throughout much of the process. In practice, this means treating all points of view and ideas brought forward in a meeting equally and ensuring that your own areas of interest or ownership are not favored over others.

There may be a stage in the process where a decision maker has to weigh the cost and ROI of possible solutions against the company roadmap though even then, ensuring that the decision made is based on merit and not personal opinion.

Empathy map #frame insights #create #design #issue analysis An empathy map is a tool to help a design team to empathize with the people they are designing for. You can make an empathy map for a group of people or for a persona. To be used after doing personas when more insights are needed.

Being a good leader means getting a team aligned, energized and focused around a common goal. In the problem solving process, strong leadership helps ensure that the process is efficient, that any conflicts are resolved and that a team is managed in the direction of success.

It’s common for managers or executives to assume this role in a problem solving workshop, though it’s important that the leader maintains impartiality and does not bulldoze the group in a particular direction. Remember that good leadership means working in service of the purpose and team and ensuring the workshop is a safe space for employees of any level to contribute. Take a look at our leadership games and activities post for more exercises and methods to help improve leadership in your organization.

Leadership Pizza #leadership #team #remote-friendly This leadership development activity offers a self-assessment framework for people to first identify what skills, attributes and attitudes they find important for effective leadership, and then assess their own development and initiate goal setting.

In the context of problem solving, mediation is important in keeping a team engaged, happy and free of conflict. When leading or facilitating a problem solving workshop, you are likely to run into differences of opinion. Depending on the nature of the problem, certain issues may be brought up that are emotive in nature.

Being an effective mediator means helping those people on either side of such a divide are heard, listen to one another and encouraged to find common ground and a resolution. Mediating skills are useful for leaders and managers in many situations and the problem solving process is no different.

Conflict Responses #hyperisland #team #issue resolution A workshop for a team to reflect on past conflicts, and use them to generate guidelines for effective conflict handling. The workshop uses the Thomas-Killman model of conflict responses to frame a reflective discussion. Use it to open up a discussion around conflict with a team.

Planning

Solving organizational problems is much more effective when following a process or problem solving model. Planning skills are vital in order to structure, deliver and follow-through on a problem solving workshop and ensure your solutions are intelligently deployed.

Planning skills include the ability to organize tasks and a team, plan and design the process and take into account any potential challenges. Taking the time to plan carefully can save time and frustration later in the process and is valuable for ensuring a team is positioned for success.

3 Action Steps #hyperisland #action #remote-friendly This is a small-scale strategic planning session that helps groups and individuals to take action toward a desired change. It is often used at the end of a workshop or programme. The group discusses and agrees on a vision, then creates some action steps that will lead them towards that vision. The scope of the challenge is also defined, through discussion of the helpful and harmful factors influencing the group.

Prioritization

As organisations grow, the scale and variation of problems they face multiplies. Your team or is likely to face numerous challenges in different areas and so having the skills to analyze and prioritize becomes very important, particularly for those in leadership roles.

A thorough problem solving process is likely to deliver multiple solutions and you may have several different problems you wish to solve simultaneously. Prioritization is the ability to measure the importance, value, and effectiveness of those possible solutions and choose which to enact and in what order. The process of prioritization is integral in ensuring the biggest challenges are addressed with the most impactful solutions.

Impact and Effort Matrix #gamestorming #decision making #action #remote-friendly In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

Project management

Some problem solving skills are utilized in a workshop or ideation phases, while others come in useful when it comes to decision making. Overseeing an entire problem solving process and ensuring its success requires strong project management skills.

While project management incorporates many of the other skills listed here, it is important to note the distinction of considering all of the factors of a project and managing them successfully. Being able to negotiate with stakeholders, manage tasks, time and people, consider costs and ROI, and tie everything together is massively helpful when going through the problem solving process.

Record keeping

Working out meaningful solutions to organizational challenges is only one part of the process. Thoughtfully documenting and keeping records of each problem solving step for future consultation is important in ensuring efficiency and meaningful change.

For example, some problems may be lower priority than others but can be revisited in the future. If the team has ideated on solutions and found some are not up to the task, record those so you can rule them out and avoiding repeating work. Keeping records of the process also helps you improve and refine your problem solving model next time around!

Personal Kanban #gamestorming #action #agile #project planning Personal Kanban is a tool for organizing your work to be more efficient and productive. It is based on agile methods and principles.

Research skills

Conducting research to support both the identification of problems and the development of appropriate solutions is important for an effective process. Knowing where to go to collect research, how to conduct research efficiently, and identifying pieces of research are relevant are all things a good researcher can do well.

In larger groups, not everyone has to demonstrate this ability in order for a problem solving workshop to be effective. That said, having people with research skills involved in the process, particularly if they have existing area knowledge, can help ensure the solutions that are developed with data that supports their intention. Remember that being able to deliver the results of research efficiently and in a way the team can easily understand is also important. The best data in the world is only as effective as how it is delivered and interpreted.

Customer experience map #ideation #concepts #research #design #issue analysis #remote-friendly Customer experience mapping is a method of documenting and visualizing the experience a customer has as they use the product or service. It also maps out their responses to their experiences. To be used when there is a solution (even in a conceptual stage) that can be analyzed.

Risk management

Managing risk is an often overlooked part of the problem solving process. Solutions are often developed with the intention of reducing exposure to risk or solving issues that create risk but sometimes, great solutions are more experimental in nature and as such, deploying them needs to be carefully considered.

Managing risk means acknowledging that there may be risks associated with more out of the box solutions or trying new things, but that this must be measured against the possible benefits and other organizational factors.

Be informed, get the right data and stakeholders in the room and you can appropriately factor risk into your decision making process.

Decisions, Decisions… #communication #decision making #thiagi #action #issue analysis When it comes to decision-making, why are some of us more prone to take risks while others are risk-averse? One explanation might be the way the decision and options were presented. This exercise, based on Kahneman and Tversky’s classic study , illustrates how the framing effect influences our judgement and our ability to make decisions . The participants are divided into two groups. Both groups are presented with the same problem and two alternative programs for solving them. The two programs both have the same consequences but are presented differently. The debriefing discussion examines how the framing of the program impacted the participant’s decision.

Team-building

No single person is as good at problem solving as a team. Building an effective team and helping them come together around a common purpose is one of the most important problem solving skills, doubly so for leaders. By bringing a team together and helping them work efficiently, you pave the way for team ownership of a problem and the development of effective solutions.

In a problem solving workshop, it can be tempting to jump right into the deep end, though taking the time to break the ice, energize the team and align them with a game or exercise will pay off over the course of the day.

Remember that you will likely go through the problem solving process multiple times over an organization’s lifespan and building a strong team culture will make future problem solving more effective. It’s also great to work with people you know, trust and have fun with. Working on team building in and out of the problem solving process is a hallmark of successful teams that can work together to solve business problems.

9 Dimensions Team Building Activity #ice breaker #teambuilding #team #remote-friendly 9 Dimensions is a powerful activity designed to build relationships and trust among team members. There are 2 variations of this icebreaker. The first version is for teams who want to get to know each other better. The second version is for teams who want to explore how they are working together as a team.

Time management

The problem solving process is designed to lead a team from identifying a problem through to delivering a solution and evaluating its effectiveness. Without effective time management skills or timeboxing of tasks, it can be easy for a team to get bogged down or be inefficient.

By using a problem solving model and carefully designing your workshop, you can allocate time efficiently and trust that the process will deliver the results you need in a good timeframe.

Time management also comes into play when it comes to rolling out solutions, particularly those that are experimental in nature. Having a clear timeframe for implementing and evaluating solutions is vital for ensuring their success and being able to pivot if necessary.

Improving your skills at problem solving is often a career-long pursuit though there are methods you can use to make the learning process more efficient and to supercharge your problem solving skillset.

Remember that the skills you need to be a great problem solver have a large overlap with those skills you need to be effective in any role. Investing time and effort to develop your active listening or critical thinking skills is valuable in any context. Here are 7 ways to improve your problem solving skills.

Share best practices

Remember that your team is an excellent source of skills, wisdom, and techniques and that you should all take advantage of one another where possible. Best practices that one team has for solving problems, conducting research or making decisions should be shared across the organization. If you have in-house staff that have done active listening training or are data analysis pros, have them lead a training session.

Your team is one of your best resources. Create space and internal processes for the sharing of skills so that you can all grow together.

Ask for help and attend training

Once you’ve figured out you have a skills gap, the next step is to take action to fill that skills gap. That might be by asking your superior for training or coaching, or liaising with team members with that skill set. You might even attend specialized training for certain skills – active listening or critical thinking, for example, are business-critical skills that are regularly offered as part of a training scheme.

Whatever method you choose, remember that taking action of some description is necessary for growth. Whether that means practicing, getting help, attending training or doing some background reading, taking active steps to improve your skills is the way to go.

Learn a process

Problem solving can be complicated, particularly when attempting to solve large problems for the first time. Using a problem solving process helps give structure to your problem solving efforts and focus on creating outcomes, rather than worrying about the format.

Tools such as the seven-step problem solving process above are effective because not only do they feature steps that will help a team solve problems, they also develop skills along the way. Each step asks for people to engage with the process using different skills and in doing so, helps the team learn and grow together. Group processes of varying complexity and purpose can also be found in the SessionLab library of facilitation techniques . Using a tried and tested process and really help ease the learning curve for both those leading such a process, as well as those undergoing the purpose.

Effective teams make decisions about where they should and shouldn’t expend additional effort. By using a problem solving process, you can focus on the things that matter, rather than stumbling towards a solution haphazardly.

Create a feedback loop

Some skills gaps are more obvious than others. It’s possible that your perception of your active listening skills differs from those of your colleagues.

It’s valuable to create a system where team members can provide feedback in an ordered and friendly manner so they can all learn from one another. Only by identifying areas of improvement can you then work to improve them.

Remember that feedback systems require oversight and consideration so that they don’t turn into a place to complain about colleagues. Design the system intelligently so that you encourage the creation of learning opportunities, rather than encouraging people to list their pet peeves.

While practice might not make perfect, it does make the problem solving process easier. If you are having trouble with critical thinking, don’t shy away from doing it. Get involved where you can and stretch those muscles as regularly as possible.

Problem solving skills come more naturally to some than to others and that’s okay. Take opportunities to get involved and see where you can practice your skills in situations outside of a workshop context. Try collaborating in other circumstances at work or conduct data analysis on your own projects. You can often develop those skills you need for problem solving simply by doing them. Get involved!

Use expert exercises and methods

Learn from the best. Our library of 700+ facilitation techniques is full of activities and methods that help develop the skills you need to be an effective problem solver. Check out our templates to see how to approach problem solving and other organizational challenges in a structured and intelligent manner.

There is no single approach to improving problem solving skills, but by using the techniques employed by others you can learn from their example and develop processes that have seen proven results.

Try new ways of thinking and change your mindset

Using tried and tested exercises that you know well can help deliver results, but you do run the risk of missing out on the learning opportunities offered by new approaches. As with the problem solving process, changing your mindset can remove blockages and be used to develop your problem solving skills.

Most teams have members with mixed skill sets and specialties. Mix people from different teams and share skills and different points of view. Teach your customer support team how to use design thinking methods or help your developers with conflict resolution techniques. Try switching perspectives with facilitation techniques like Flip It! or by using new problem solving methodologies or models. Give design thinking, liberating structures or lego serious play a try if you want to try a new approach. You will find that framing problems in new ways and using existing skills in new contexts can be hugely useful for personal development and improving your skillset. It’s also a lot of fun to try new things. Give it a go!

Encountering business challenges and needing to find appropriate solutions is not unique to your organization. Lots of very smart people have developed methods, theories and approaches to help develop problem solving skills and create effective solutions. Learn from them!

Books like The Art of Thinking Clearly , Think Smarter, or Thinking Fast, Thinking Slow are great places to start, though it’s also worth looking at blogs related to organizations facing similar problems to yours, or browsing for success stories. Seeing how Dropbox massively increased growth and working backward can help you see the skills or approach you might be lacking to solve that same problem. Learning from others by reading their stories or approaches can be time-consuming but ultimately rewarding.

A tired, distracted mind is not in the best position to learn new skills. It can be tempted to burn the candle at both ends and develop problem solving skills outside of work. Absolutely use your time effectively and take opportunities for self-improvement, though remember that rest is hugely important and that without letting your brain rest, you cannot be at your most effective.

Creating distance between yourself and the problem you might be facing can also be useful. By letting an idea sit, you can find that a better one presents itself or you can develop it further. Take regular breaks when working and create a space for downtime. Remember that working smarter is preferable to working harder and that self-care is important for any effective learning or improvement process.

Want to design better group processes?

Over to you

Now we’ve explored some of the key problem solving skills and the problem solving steps necessary for an effective process, you’re ready to begin developing more effective solutions and leading problem solving workshops.

Need more inspiration? Check out our post on problem solving activities you can use when guiding a group towards a great solution in your next workshop or meeting. Have questions? Did you have a great problem solving technique you use with your team? Get in touch in the comments below. We’d love to chat!

James Smart is Head of Content at SessionLab. He’s also a creative facilitator who has run workshops and designed courses for establishments like the National Centre for Writing, UK. He especially enjoys working with young people and empowering others in their creative practice.

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Mastering IELTS Speaking: How to Describe a Person with a Unique Problem-Solving Approach

In the IELTS Speaking test, candidates often encounter questions that require them to describe people with specific qualities or attributes. One such topic that has gained popularity in recent years is “ Describe A Person Who Has A Unique Approach To Problem-solving .” This article will provide you with valuable insights and strategies to excel in this particular speaking task, helping you achieve a higher band score in your IELTS Speaking test.

Table of Contents

1 Understanding the Topic
2 Part 1: Introduction and Interview
3.1 Additional questions the examiner might ask:
4 Part 3: Two-way Discussion
5 Key Vocabulary and Phrases for High Scores
6 Tips from an IELTS Speaking Examiner

Understanding the Topic

Before we delve into the specifics of answering this question, it’s essential to understand what the examiner is looking for when they ask you to describe a person with a unique problem-solving approach. They want to assess your ability to:

Articulate your thoughts clearly and coherently
Use a range of vocabulary related to problem-solving and personal qualities
Demonstrate your understanding of what makes an approach “unique”
Provide specific examples to support your description

Part 1: Introduction and Interview

In Part 1 of the IELTS Speaking test, the examiner may ask you some general questions related to problem-solving. Here are a few examples:

Do you enjoy solving problems?
What kind of problems do you usually face in your daily life?
How do you typically approach solving a problem?

Let’s focus on answering the first question:

Examiner: Do you enjoy solving problems?

Candidate: Absolutely! I find problem-solving to be both challenging and rewarding . There’s a certain thrill in tackling a complex issue and coming up with an effective solution. It’s like solving a puzzle, and the sense of accomplishment when you finally crack it is incredibly satisfying. Plus, I believe that honing my problem-solving skills is crucial for personal and professional growth, so I always welcome opportunities to face new challenges.

In this response, the candidate demonstrates enthusiasm, uses relevant vocabulary, and provides a brief explanation of why they enjoy problem-solving. The bolded phrases are particularly effective in showcasing language proficiency and thoughtful reflection.

Part 2: Long Turn (Cue Card)

In Part 2, you might receive a cue card related to describing a person with a unique problem-solving approach. Here’s an example:

Describe a person who has a unique approach to problem-solving

You should say:

Who this person is
What kind of problems they solve
How their approach is unique
And explain why you think their approach is effective

Here’s a sample answer:

I’d like to talk about my former university professor, Dr. Emily Chen, who has a truly unique approach to problem-solving. Dr. Chen is an innovative thinker** in the field of environmental science, and she primarily tackles complex issues related to climate change and sustainable development.

What sets Dr. Chen apart is her unconventional methodology . Instead of approaching environmental problems from a purely scientific standpoint, she integrates principles from diverse disciplines such as psychology, economics, and even art. This interdisciplinary approach allows her to view challenges from multiple angles and come up with holistic solutions .

One of the most fascinating aspects of her problem-solving technique is how she incorporates creative visualization . For instance, when addressing the issue of plastic pollution in oceans, she organized a workshop where participants created art installations using recycled plastic. This not only raised awareness but also sparked innovative ideas for reducing plastic waste.

I believe her approach is particularly effective because it engages people on an emotional level , making abstract environmental concepts more tangible and relatable. By bridging the gap between science and creativity , Dr. Chen manages to inspire action and foster collaboration among diverse groups of people.

Her unique problem-solving style has led to several groundbreaking initiatives , including a city-wide program that reduced carbon emissions by 30% in just two years. It’s this ability to think outside the box and bring together seemingly unrelated ideas that makes Dr. Chen’s approach so effective and inspiring.**

This response effectively addresses all the points in the cue card while showcasing a range of vocabulary and complex sentence structures. The bolded phrases highlight key language that demonstrates a high level of English proficiency.

Additional questions the examiner might ask:

How did you first come to know about Dr. Chen’s unique approach?
Can you give another example of a problem she solved using her method?

Sample answer for question 1:

I first learned about Dr. Chen’s unique approach when I attended one of her guest lectures at my university. She was presenting her latest research on urban sustainability, and I was immediately captivated by how she wove together concepts from urban planning, behavioral psychology, and ecological design. It was eye-opening to see how she could draw connections between such diverse fields and apply them to real-world environmental challenges.

Part 3: Two-way Discussion

In Part 3, the examiner will ask more abstract questions related to the topic. Here are some possible questions and sample answers:

Examiner: Do you think schools should teach problem-solving skills?

Candidate: Absolutely. I believe that cultivating problem-solving skills should be a fundamental component of education. In today’s rapidly changing world, students need to be equipped with the ability to tackle complex challenges and adapt to new situations . By integrating problem-solving exercises into various subjects, schools can help students develop critical thinking , creativity , and resilience – skills that are invaluable in both personal and professional life.

Moreover, teaching problem-solving skills can help students become more self-reliant and confident in their abilities. It encourages a growth mindset , where challenges are seen as opportunities for learning rather than insurmountable obstacles. This approach to education would better prepare students for the realities of the modern workforce, where innovative thinking and adaptability are highly prized.

Examiner: How has technology changed the way we solve problems?

Candidate: Technology has revolutionized problem-solving in numerous ways. Firstly, it has given us access to vast amounts of information at our fingertips, allowing us to research and analyze issues more thoroughly and quickly than ever before. This wealth of data enables us to make more informed decisions and consider a wider range of potential solutions.

Secondly, technology has facilitated collaboration on a global scale. Through online platforms and communication tools, experts from different fields and geographical locations can pool their knowledge and work together to tackle complex problems. This cross-pollination of ideas often leads to more innovative and comprehensive solutions .

Additionally, advanced technologies like artificial intelligence and machine learning are enhancing our problem-solving capabilities by identifying patterns and processing data at speeds far beyond human capacity. This allows us to address issues that were previously too complex or time-consuming to solve manually.

However, it’s important to note that while technology has greatly augmented our problem-solving abilities, it has also introduced new challenges , such as information overload and the need for digital literacy. Striking a balance between leveraging technology and maintaining human intuition and creativity is crucial for effective problem-solving in the modern era.

Key Vocabulary and Phrases for High Scores

To achieve a high band score in IELTS Speaking, it’s essential to use a range of sophisticated vocabulary and expressions. Here are some key terms related to problem-solving and unique approaches:

Innovative thinker /ˈɪnəveɪtɪv ˈθɪŋkər/ (noun): Someone who comes up with new and creative ideas. Example: Her reputation as an innovative thinker has led to numerous speaking engagements.

Think outside the box /θɪŋk aʊtˈsaɪd ðə bɒks/ (idiom): To think creatively and unconventionally. Example: To solve this complex issue, we need to think outside the box.

Holistic approach /həˈlɪstɪk əˈprəʊtʃ/ (noun): Considering all aspects of a problem or situation. Example: He takes a holistic approach to health, focusing on both physical and mental well-being.

Paradigm shift /ˈpærədaɪm ʃɪft/ (noun): A fundamental change in approach or underlying assumptions. Example: Her method represents a paradigm shift in how we approach environmental conservation.

Lateral thinking /ˈlætərəl ˈθɪŋkɪŋ/ (noun): Solving problems through an indirect and creative approach. Example: Lateral thinking is often necessary when traditional methods fail to yield results.

Tips from an IELTS Speaking Examiner

As an experienced IELTS Speaking Examiner, I recommend the following strategies to excel in this task:

Practice describing people : Regularly practice describing people you know who have unique qualities or approaches. This will help you build a repertoire of relevant vocabulary and expressions.

Use specific examples : Always support your statements with concrete examples. This adds credibility to your description and demonstrates your ability to elaborate on ideas.

Focus on the ‘unique’ aspect : When describing the person’s problem-solving approach, emphasize what makes it different or innovative. This shows that you understand the key elements of the task.

Structure your response : Organize your thoughts logically, addressing each point in the cue card systematically. This helps ensure you cover all aspects of the question.

Expand your vocabulary : Continuously work on expanding your vocabulary related to personal qualities, problem-solving, and innovative thinking. This will allow you to express your ideas more precisely and eloquently.

By following these strategies and incorporating the sample answers and vocabulary provided, you’ll be well-equipped to tackle the “Describe a person who has a unique approach to problem-solving” topic in your IELTS Speaking test. Remember, practice is key to improving your speaking skills and achieving a high band score.

For more tips on acing your IELTS Speaking test, check out our articles on describing a time when you had to lead a group through a challenging situation and describing a person who has a unique approach to life .

Mastering IELTS Speaking: How to Describe a Time When You Had to Work with Limited Resources

Disciplined person following daily routine

How to Describe a Disciplined Person: Ace Your IELTS Speaking Test

How to Effectively Stand Up for Your Beliefs in IELTS Speaking

The process of problem posing: development of a descriptive phase model of problem posing

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Published: 27 December 2021
Volume 110 , pages 251–269, ( 2022 )

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Lukas Baumanns ORCID: orcid.org/0000-0002-6697-3994 1 &
Benjamin Rott ORCID: orcid.org/0000-0002-8113-1584 1

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The aim of this study is to develop a descriptive phase model for problem-posing activities based on structured situations. For this purpose, 36 task-based interviews with pre-service primary and secondary mathematics teachers working in pairs who were given two structured problem-posing situations were conducted. Through an inductive-deductive category development, five types of activities (situation analysis, variation, generation, problem-solving, evaluation) were identified. These activities were coded in so-called episodes, allowing time-covering analyses of the observed processes. Recurring transitions between these episodes were observed, through which a descriptive phase model was derived. In addition, coding of the developed episode types was validated for its interrater agreement.

A model for problem creation: implications for teacher training

On understanding mathematical problem-posing processes

Problem Posing as Reformulation and Sense-Making Within Problem Solving

Avoid common mistakes on your manuscript.

1 Introduction

In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi . (Cantor, 1867 , p. 26) Transl.: In mathematics, the art of posing a question is of greater value than solving it.

In his statement, Cantor emphasizes the importance of the ability to pose substantial questions within mathematics. In fact, problem posing is considered a central activity of mathematics (Hadamard, 1945; Halmos, 1980 ), and at the latest since the 1980s (Brown & Walter, 1983 ; Butts, 1980 ; Kilpatrick, 1987 ), it is being investigated with growing interest by mathematics education researchers. Since the 1990s, it has been widely used to identify or assess mathematical creativity and abilities (Silver, 1994 , 1997 ; Singer & Voica, 2015 ; Van Harpen & Sriraman, 2013 ; Yuan & Sriraman, 2011 ). Silver ( 1997 , p. 76) emphasizes that to grasp such constructs, both products and processes of problem-posing activities can be considered. However, a strong product orientation within research on problem posing is noticeable (Bonotto, 2013 ; Singer et al., 2017 ; Van Harpen & Sriraman, 2013 ); that is, studies aiming to assess mathematical creativity, for example, often focus on the posed problems rather than the processes that led to them. This is noteworthy, since processes are central to educational research. As Freudenthal ( 1991 ) states:

[T]he use of and the emphasis on processes is a didactical principle . Indeed, didactics itself is concerned with processes. Most educational research, however, and almost all of it that is based on or related to empirical evidence, focuses on states (or time sequences of states when education is to be viewed as development). States are products of previous processes . As a matter of fact, products of learning are more easily accessible to observation and analysis than are learning processes which, on the one hand, explains why researchers prefer to deal with states (or sequences of states), and on the other hand why much of this educational research is didactically pointless. (p. 87, emphases in original)

Although there are studies considering problem-posing processes (Headrick et al., 2020 ; Ponte & Henriques, 2013 ), general knowledge about learners’ problem-posing processes remains limited (Cai & Leikin, 2020 ). Only a few studies are dedicated to the development of a phase model for problem posing (Cruz, 2006 ; Pelczer & Gamboa, 2009 ). Those models still hold the potential for sufficient generalization and validation. This knowledge could help to develop a more sophisticated process-oriented perspective on problem posing. The few studies that examine the general process of problem posing (Koichu & Kontorovich, 2013 ; Patáková, 2014 ; Pelczer & Rodríguez, 2011 ) may benefit from a validated phase model. Such a model may also be useful for the effective educational use of problem posing in the classroom. This study aims to develop a valid and reliable category system that allows analyzing problem-posing processes. These kinds of conceptual frameworks play a central role in mathematics education research as they enable a better understanding of thinking processes (Lester, 2005 ; Schoenfeld, 2000 ).

2 Theoretical background

2.1 problem posing.

There are two widespread definitions of problem posing which are used or referred to in most studies on the topic. As a first definition, Silver ( 1994 , p. 19) describes problem posing as the generation of new problems and reformulation of given problems. Silver continues that both activities can occur before, during, or after a problem-solving process. As a second definition, Stoyanova and Ellerton ( 1996 , p. 218) refer to problem posing as the “process by which, on the basis of mathematical experience, students construct personal interpretations of concrete situations and formulate them as meaningful mathematical problems.” In the following, we adopt the definition of Silver ( 1994 ) as the differentiation between the activities of generation and reformulation is beneficial for identifying different activities in problem-posing processes. However, both definitions are not disjunctive or contradictory but describe equivalent activities.

In both definitions, the term problem is used for any kind of mathematical task, whether it is a routine or a non-routine problem (Pólya, 1966 ). For the former, “one has ready access to a solution schema” (Schoenfeld, 1985b , p. 74), and for the latter, one has no access to a solution schema. Thus, problem posing can lead to any kind of task on the spectrum between routine and non-routine problems (Baumanns & Rott, 2019; Baumanns & Rott, 2021a ).

Stoyanova and Ellerton ( 1996 ) distinguish between free, semi-structured, and structured problem-posing situations depending on the degree of structure. A situation is an ill-structured problem in the sense that its goal cannot be determined by all given elements and relationships (Stoyanova, 1997 ). Because this study focuses on structured situations and Baumanns and Rott ( 2021a ) encountered difficulties in distinguishing free and semi-structured situations, in this article, we distinguish between unstructured and structured situations. Unstructured situations form a spectrum of situations without an initial problem. The given information of these situations reaches from nearly none (see Table 1 , situation 1) to open situations with numerous given information, the structure of which must be explored by using mathematical knowledge and mathematical concepts (see Table 1 , situation 2). In structured situations, people are asked to pose further problems based on a specific problem, for example, by varying its conditions (see Table 1 , situation 3). The phase model developed in this article aims to describe problem-posing activities that are induced by situations like those in Table 1 . In particular, the model is developed using structured situations.

2.2 Process of problem posing—state of research

Because products may be more accessible by analysis than processes (Freudenthal, 1991 ), most problem-posing studies focus on posed problems (Bicer et al., 2020 ; Van Harpen & Presmeg, 2013 ; Yuan & Sriraman, 2011 ). However, consideration of the processes increases in recent studies (Cai & Leikin, 2020 ; Crespo & Harper, 2020 ; Headrick et al., 2020 ; Koichu & Kontorovich, 2013 ; Patáková, 2014 ; Pelczer & Rodríguez, 2011 ). Ponte and Henriques ( 2013 ), for example, examine the problem-posing process in investigation tasks among university students and found that problem posing and problem-solving complement each other in generalizing or specifying conjectures to obtain more general knowledge about the mathematics contents. Christou et al. ( 2005 ) describe four thinking processes that occur within problem posing, namely editing, selecting, comprehending/organizing, and translating quantitative information. They found the most able students are characterized through editing and selecting processes. However, compared to the present study, these activities do not tend to describe problem-posing processes by phases. Instead, Christou et al. ( 2005 ) intend to characterize thinking processes in problem posing. Cifarelli and Cai ( 2005 ) include problem posing in their model to describe the structure of mathematical exploration in open-ended problem situations. They identify a recursive process in which reflection on a problem’s solutions serves as the source of new problems.

The studies cited above differ from the present study as follows: They describe and analyze only individual processes, they describe the problem-posing process in terms of thinking processes rather than phases, or they consider problem-posing processes as a sub-phase of a superordinate process. However, there is a lack of studies that attempt to derive a general, descriptive phase model of observed problem-posing processes themselves from numerous processes. For problem-solving research, the analysis of processes through phase models has been established at least since Pólya’s ( 1945 ) and Schoenfeld’s ( 1985b ) seminal works. While their models are normative , which means they function as advice on how to solve problems, newer empirical studies on the process of problem-solving develop and investigate descriptive models, which means they portray how problems are actually solved by participants (Artzt & Armour-Thomas, 1992 ; Rott et al., 2021 ; Yimer & Ellerton, 2009 ). This study also focuses on descriptive models.

Some researchers interpret problem posing as a problem-solving activity (Arıkan & Ünal, 2015 ; Kontorovich et al., 2012 ; Silver, 1995 ), and there are several established models of problem-solving processes (e.g., Mason et al., 1982 ; Pólya, 1945 ). Therefore, it is a reasonable question whether a separate phase model for problem posing is needed. From the observations of problem-solving and problem-posing activities within the present study, we share the argument by Pelczer and Gamboa ( 2009 ) that the cognitive processes involved in problem posing are of their own nature and cannot be adequately described by the phase models of problem solving. For problem posing, Cai et al. ( 2015 ) state, “there is not yet a general problem-posing analogue to well-established general frameworks for problem solving such as Polya’s (1957) four steps” (p. 14).

To find existing research on this topic, we conducted a systematic literature review (Baumanns & Rott, 2021a , 2021b ). This review encompassed articles from high-ranked journals of mathematics education, the Web of Science, PME proceedings, the 2013 and 2020 special issues in Educational Studies of Mathematics , the 2020 special issue in International Journal of Educational Research , and two edited books on problem posing (Felmer et al., 2016 ; Singer et al., 2015 ). From all reviewed articles, three were dedicated to the development of general phases in problem posing similarly to the present study.

Cruz ( 2006 ) postulates a phase model based on a training program for teachers (see Fig. 1 ). For this reason, this phase model is preceded by educative needs and goals. Once a concrete teaching goal has been set (1), the episode type of problem formulating begins (2). This episode has a problem as its output which is then solved (3). If it cannot be solved, the problem may have to be reformulated (4). A solvable problem is further developed in the episode type problem improving (5). The complexity of the problem is adapted to the learning group and compared with the goal (6 and 7). If the comparison shows that the problem is not suitable, either further changes are made to the task (8) or the task is rejected as unsuitable.

Phase model of problem posing by Cruz ( 2006 )

Pelczer and Gamboa ( 2009 ) distinguish five phases— setup , transformation , formulation , evaluation , and final assessment— based on the analysis of problem-posing processes in unstructured situations. The setup includes the definition of the mathematical context of a situation and the reflection on the knowledge needed to understand the situation. This assessment serves as a starting point for the subsequent process. During the transformation , the conditions of a problem are analyzed, and possibilities for modification are identified, reflected, and executed. In the formulation , all activities related to the formulation of a task are summarized. This includes the consideration of different possible formulations of the problem as well as an evaluation of these formulations. In the evaluation , a posed problem is assessed in terms of various aspects, for example, whether it fulfills the initial conditions or further modifications are needed. In the final assessment , the process of posing a problem is reflected upon, and the problem itself is evaluated, for example, in terms of difficulty and interest. In their study, Pelczer and Gamboa ( 2009 ) compare experts’ and novices’ problem-posing processes, identifying different trajectories, that is, transitions between the stages. While experts more often go through recursive processes, processes of novices are more linear and often occur without transformation and final assessment.

Koichu and Kontorovich ( 2013 ) developed four stages, observed in the context of two successful problem-posing activities: (1) In the warming-up phase, typical problems spontaneously associated with the given situation are posed that serve as a starting point. (2) In the phase searching for an interesting mathematical phenomenon , participants concentrate on selected aspects of the given task to identify interesting aspects that can be used for forthcoming problems. (3) Since the intention is to develop interesting problem formulations, in the phase hiding the problem-posing process in the problem formulation , the posers try to disguise to the potential solvers in which way the task was created. (4) Finally, in the reviewing phase, the posers evaluate the problems based on individual criteria such as the degree of difficulty or appropriateness for a specific target group.

In general, Cruz’ ( 2006 ) phase model does not allow for sufficient generalization to processes of sample groups that do not pursue school learning goals such as students or mathematicians. The model by Pelczer and Gamboa ( 2009 ) has the potential to verify the validity by checking objective coding. The stages by Koichu and Kontorovich ( 2013 ) are developed on a small sample of two people and therefore need to be tested for applicability to larger sample groups. All these potentials will be addressed in this article. In addition, although the models presented have certain similarities, they also show numerous characteristic differences. In comparison, phase models for problem-solving (Artzt & Armour-Thomas, 1992 ; Pólya, 1945 ; Rott et al., 2021 ; Schoenfeld, 1985b ; Yimer & Ellerton, 2009 ) share a very similar core structure. Thus, there is a conceptual and empirical need for a generally applicable model for problem-posing research.

The need for developing a phase model for problem posing is, furthermore, based on our general observation that the quality of the posed problems did not always match the quality of the observed activity. In our opinion, it is therefore not enough to consider only the products when, for example, problem posing is used to assess mathematical creativity. Furthermore, developing a process-oriented framework serves as research for discussing and analyzing these processes (Fernandez et al., 1994 , p. 196).

2.3 Research questions

The research goal of this study is to develop a descriptive phase model for problem-posing activities based on structured situations. The lack of phase models constitutes a desideratum from which the following research questions emerge:

Which recurring and distinguishable activities can be identified when dealing with structured problem-posing situations?

What is the general structure (i.e., sequence of distinguishable activities) of the observed processes from which a descriptive phase model may be derived?

The goal of these research questions is to develop a descriptive phase model that allows analyzing problem-posing processes. To evaluate the quality of this model, we draw on the criteria by Schoenfeld ( 2000 ) that can be used for evaluating models in mathematics education. As this type of coding is highly inferential (Rott et al., 2021 ; Schoenfeld, 1985b ), special emphasis is given to interrater agreement.

3 The study

3.1 data collection.

The present study is a generative study that aims to “generate new observation categories and new elements of a theoretical model in the form of descriptions of mental structures or processes that explain the data” (Clement, 2000 , p. 557). For such studies, a less structured, qualitative approach is appropriate that is open to unexpected findings (Döring & Bortz, 2016 , p. 192), such as task-based interviews. Task-based interviews have particularly been used in problem-solving research to gain insights into the cognitive processes of participants (Konrad, 2010 , p. 482). The interviews were conducted in pairs to create a more natural communication situation and eliminate the constructed pressure to produce something mathematical for the researcher (Schoenfeld, 1985a , p. 178). Johnson and Johnson ( 1999 ) also underline that cooperative learning groups such as pairs are “windows into students’ minds” (p. 213). For this reason, the interviewer avoided intervening in the interaction process.

The interviews were conducted with 64 pre-service primary and secondary mathematics teachers (PST). The PSTs worked in pairs on one of two structured problem-posing situations, either (A) Nim game or (B) Number pyramid, which are presented in Table 2 . The participants were informed that both problem solving and problem posing were central. After the initial problem solving, both situations stated: “Based on this task, pose as many mathematical tasks as possible.” This open and restriction-free question should stimulate a creative process. A common question of understanding from participants was, using the example of situation (A), whether they should now pose further Nim games or were also allowed to depart from them. This decision was left to the PSTs’ creativity.

In total, 15 processes of situation (A) and 17 processes of situation (B), ranging from 9 to 25 min, have been recorded and analyzed. The processes ended when no ideas for further problems emerged from the participants. In total, 7 h and 46 min of video material were recorded and analyzed. Thus, the processes had an average length of 14.5 min. Four pairs of PSTs each were in the same room under authentic university seminar conditions. A camera was positioned opposite the pairs capturing all the participants’ actions. To accustom them to natural communication in front of the camera, short puzzles were performed before problem posing.

3.2 Data analysis

For data analysis, we adapted Schoenfeld’s ( 1985b ) verbal protocol analysis , originally used to analyze problem-solving processes. This method is an event-based sampling. Compared to time-based sampling, the processes are not divided into fixed time segments (e.g., 30 s), which are then coded. Instead, new codes are set when the participants’ behavior changes. This method has two steps: At first, the recorded interviews are segmented into “macroscopic chunks of consistent behavior” (Schoenfeld, 1985b , p. 292) that are called episodes in which “an individual or a problem-solving group is engaged in one large task [...] or closely related body of tasks in the service of the same goal” (Schoenfeld, 1985b , p. 292). In a second step, the episodes are then characterized in terms of content.

To answer the first research question, verbal protocol analyses were employed in terms of inductive category development (Mayring, 2014 , pp. 79–87), meaning that the episode types were developed data-derived. The descriptions of the episode types were additionally concretized in a theory-based manner. For that, the above-mentioned conceptual and empirical findings of problem-posing research (Cruz, 2006 ; Pelczer & Gamboa, 2009 ; Silver, 1994 ), as well as findings of research on phase models in problem solving (Pólya, 1945 ; Schoenfeld, 1985b ), were used. This procedure aims to develop exclusive and exhaustive codes (Cohen, 1960 ), that is, episode types, that can be assigned to the observed problem-posing processes.

To answer the second research question, recurring sequences of the episode types were identified to develop a general phase model. Both general sequences in the observed processes, as well as conceptual insights about problem-posing activities in general, were considered. To analyze the interrater agreement, an independent second coder was trained. At first, the second coder was given the coding manual and a process to code without further comment. For this first coding, cases of doubt were discussed within 2 h of training. After this training, the second coder analyzed about 2 h and 23 min of the total video material of 7 h and 46 min which means 10 randomly chosen processes out of 32. Thus, the second coder analyzed about 30.7% of the total video material. Finally, cases of doubt of coding were discussed via consensual validation. These codings were used to calculate the interrater agreement to the author’s coding.

The interrater agreement was calculated with the EasyDIAg algorithm by Holle and Rein ( 2015 ). EasyDIAg provides an algorithm that converts two codes of an event-based sampling data set into an agreement table from which Cohen’s kappa (Cohen, 1960 ) is calculated through an iterative proportional fitting algorithm. Furthermore, in contrast to the classical Cohen’s kappa, EasyDIAg provides an interrater agreement score for each value of a category. EasyDIAg considers raters’ agreement on segmentation and categorization as well as the temporal overlap of the raters’ annotations. This makes this algorithm particularly suitable for assessing the interrater agreement of the event-based sampling data set at hand. For the agreement, we used an overlap criterion of 60% as suggested by Holle and Rein ( 2015 ). In the online supplement, we provide an example analysis of a process that was coded by the authors and the second rater followed by the calculation of the interrater agreement in this manner.

First, to retrace the inductive-deductive category development, the problem-posing process of the Nim game by Theresa and Ugur will be described in order to refer back to it when describing the developed episode types. The individual episodes are described without labelling them. The given periods indicate the minutes and seconds ( mm:ss ) of the respective episodes. The recorded time starts with the first attempt at posing problems after the initial problem has been solved. Compared to other participants, Theresa and Ugur get the solution of the Nim game quickly and without assistance.

Episode 1 (00:00–00:49): Theresa and Ugur first read the task that should initiate the problem posing. Ugur considers whether new tasks should now be posed in relation to the solution strategy of working backwards. Theresa considers whether the stones should be the focus of new tasks. Afterward, both reflect again on their solution strategy and consider to what extent they can use it for new tasks.

Episode 2 (00:49–02:14): Then other games like Connect Four or Tic-tac-toe , which may have a winning strategy similar to the Nim game, are collected.

Episode 3 (02:14–05:50): Both participants want to figure out whether there is a winning strategy for Tic-tac-toe. After about 3 min, they assume that an optimal game always results in a draw. They return to the Nim game and ponder whether player B also has a chance to win safely. They conclude that player B can only win if player A does not make the first move according to the winning strategy.

Episode 4 (05:50–07:43): They pose the task of how many stones are necessary for player B to win safely. Afterward, the text of the task is formulated. They also ask how many moves player A needs in order to win.

Episode 5 (07:43–09:03): The last-mentioned question of episode 4 is solved and also generalized. Ugur says, you find the number of moves of player A by going from the number of stones to the next higher number divisible by three, and then dividing this number by three.

Episode 6 (09:03–09:44): Ugur suggests increasing the number of stones that can be removed from the table. Specifically, he suggests that one to three stones can be removed. Meanwhile, Theresa writes down these ideas.

Episode 7 (09:44–10:32): Theresa writes down the previously posed problems without working on the content of the formulations.

Episode 8 (10:32–13:48): Both play the variation of the Nim game raised in episode 6. They express that they want to develop a winning strategy for this variation. They quickly realize that player B can safely win the game since multiples of four are now winning numbers and the 20 stones that are on the table at the beginning are already divisible by four. They validate this strategy afterward. At the last minute, the newly posed variation is also evaluated as exciting.

Episode 9 (13:48–14:13): Ugur wants to generalize the game further and poses the task of how to win when the players can remove one to n stones. Theresa asks Ugur if his goal is a general formula.

Episode 10 (14:13–15:48): This task is then solved by Ugur by transferring the structure of the solution of the initial problem to the generalization. Ugur formulates that if you are allowed to remove one to n − 1 stones, the player who has the turn must bring the number of stones to n by his turn to win safely.

Episode 11 (15:48–16:42): Subsequently, both work on a suitable formulation for this generalized task.

Episode 12 (16:42–18:28): Theresa notes that solving the initial problem is challenging and therefore suggests providing help for pupils. Theresa suggests that it might help when the pupils first develop a winning strategy for the simple case that the players can only remove one stone. Ugur suggests further help cards which can be requested by the pupils themselves if they get stuck.

Episode 13 (18:28–19:50): Theresa wants to focus on new tasks again. They move away from the initial problem and use the stones to create an iconic representation of the triangular numbers (1, 3, 6, ...). They formulate the task to find a general formula to calculate the n -th triangular number.

Episode 14 (19:50–21:33): Theresa puts the stones in rows of three so that the structure that leads to the winning strategy is more visible. She evaluates this presentation by emphasizing the usefulness of this method for extensions of the Nim game with more than 20 stones on the table. The process comes to an end as Theresa and Ugur, when asked by the interviewer, agree not to generate any more ideas.

4.1 Category development of episode types in problem posing

Using the described evaluation method, five episode categories were developed which allow the observed processes to be described in a time-covering manner. These episode categories are situation analysis , variation , generation , problem-solving , and evaluation . In the following, the developed categories of episode types are described. The episodes of the process by Theresa and Ugur (T&U) described above are assigned to these episode types for a better comprehension of the episode types. In addition, we provide further anchor examples in the online supplement . Subsequently, indications are given for coding the individual categories. Finally, the categories are discussed regarding the state of research.

4.1.1 Situation analysis

Description.

During the situation analysis , the posers capture single or multiple conditions of the initial task. They usually recognize which conditions are suitable and to what extent, to create a new task by variation (changing or omitting single or multiple conditions) or generation (constructing single or multiple new conditions). In addition, the subsequent investigation of the initial task’s solution is summarized in this episode. This also includes the creation of clues or supporting tasks that lead to the solution of the initial task.

In the process of T&U, episode 1 is coded as situation analysis as the participants still reflect on their solution strategy. Also, episode 12 is coded as situation analysis because both PSTs try to come up with ideas on how to support students with solving the initial problem. A further example of other participants who capture the conditions of the initial problem can be found in the online supplement .

Coding instructions

It is not always clear when the posers are engaged in reading (see non-content-related episodes below) or have already moved on to situation analysis . Simultaneous coding is possible here. The creation of supporting tasks, which are supposed to assist in solving the initial problem, is interpreted as an analytical examination of the situation and is therefore coded as situation analysis .

4.1.2 Variation

During variation , single or multiple conditions of the initial task or a task previously posed in the process are changed or omitted. No additional conditions are constructed. In addition, writing down and formulating the respective task is included under this episode.

In the process of T&U, episodes 4, 6, 9, and 11 are coded as variation . In episode 6, for example, Ugur varies one specific rule of the Nim game and states that the players are now allowed to remove one to three stones from the table. In episode 9, this is further generalized by variation.

For the identification of variation , the What-If-Not-strategy by Brown and Walter ( 2005 ) should be used. The first step of this strategy is intended to extract the conditions of a problem. The Nim game, for example, has at least the following five conditions: (1) 20 stones, (2) two players, (3) alternating moves, (4) one or two stones are removed, and (5) whoever empties the table wins. This analysis should be done before coding. Omitting or varying these analyzed conditions will be coded as variation . Also, omitting or varying conditions of a previously posed problem is coded as variation .

4.1.3 Generation

During generation, tasks are raised by constructing new conditions to the given initial task or a task previously posed in the process. Due to the possible change in the task structure, posers sometimes explain the new task. In addition, writing down and formulating the respective task is summarized under this episode type. Also, free associations, in which tasks similar to the initial task are reminded, are coded as generation .

In the process of T&U, episodes 2 and 13 are coded as generation . In episode 13, for example, they move further away from the Nim game and use the stones to ask questions about dot patterns.

The episode types variation and generation are not always clearly distinguishable from each other. Although the coding focuses on the activity of the poser and not on the emerged task, it can help to examine the characteristics of a task resulting from variation or generation . In the case of a varied task, the question or the solution structure often remains unchanged. In the case of a generated task, there is usually a fundamentally different task whose solution often requires different strategies.

4.1.4 Problem solving

Problem solving describes the activity in which the posers solve a task that they have previously posed. If a non-routine problem has been posed, the respondents go through a shortened problem-solving process in which the phases of devising and carrying out the plan (Pólya, 1945 ) are the main focus. In some cases, the posers omit to carry out the plan if the plan already provides sufficient information on the solvability and complexity of the posed problem. If a routine problem has been posed, the solution is usually not explained, since the method of solution is known. However, longer phases of solving routine tasks are also coded as problem solving.

In the process of T&U, episodes 3, 5, 8, and 10 are coded as problem solving as the participants are engaged in solving their posed problems.

Although solving a routine problem should be differentiated from solving a non-routine problem, both activities are labelled with the same code. However, the commentary of the coding should specify whether an episode is an activity of solving a routine or a non-routine problem.

4.1.5 Evaluation

In the evaluation , the posers assess the posed tasks based on individually defined criteria. In the processes observed, posers asked whether the posed problem is solvable, well-defined, similar to the initial task, appropriate for a specific target group, or interesting for themselves to solve. On the basis of this evaluation, the posed task is then accepted or rejected.

In the process of T&U, episode 14 is coded as evaluation , and in episode 8, there is a simultaneous coding of problem-solving and evaluation . In episode 8, for example, the participants are initially engaged in problem-solving. Towards the end of this episode, they both assess their posed problem based on their interest in solving it.

Often, evaluative statements are made about the course of an episode of problem-solving , since the criteria for the evaluation of a posed problem (e.g., solvability or interest) are based on sufficient knowledge about the solution of the posed problem. In such cases, the episode types of problem-solving and evaluation cannot be separated empirically, which is why simultaneous coding is permitted. The criterion for this simultaneous coding is that during an episode of problem-solving , an evaluative statement must come within a 30-s window for a simultaneous coding to be made. For example, if at least one evaluative statement falls during the first 30 s of a problem-solving episode, both types of episodes are coded simultaneously. If at least one evaluative statement also falls within the following 30 s of problem-solving , both episode types are again encoded simultaneously.

4.1.6 Non-content-related episode types

When participants, for example, ran out of ideas or became distracted during the interview, they engage in the following non-content-related activities. Such activities were also identified in descriptive models of problem solving (Rott et al., 2021 ). In the process of T&U, episode 7 was coded as non-content-related episode .

The episode of reading consists of reading the situation text as well as a shorter exchange about what has been read to make sure that the text is understood. Since the participants have usually already solved the initial task of the situation, the reading takes place rather in between.

In the episode of writing, posers write down the text of a problem they have already worked out orally. Also, the posers write down the solution of a previously posed problem. Writing is only coded if no solution or problem formulation is being worked on in terms of content (e.g., specify the problem text).

Organization

Organization includes all activities in which the poser is working on the situation, but where no content-related work is apparent. This includes, for example, the lengthy production of drawings.

The episode digression is encoded when the posers are not engaged with the situation. This may include informal conversations with the other person about topics that are not related to the task (e.g., weekend activities) or looking out of the window for a long time.

All episodes that cannot be assigned to any other episode type are coded as other.

4.1.7 Discussion

To provide a theoretical justification of the data-driven episode types of problem posing, we want to connect the five episode types with the presented state of research on problem-posing phase models.

Situation analysis

In Pelczer’s and Gamboa’s ( 2009 ) phase model, we find aspects of situation analysis in their transformation stage. One sub-process of this transformation stage is the analysis of the problem’s characteristics. Terminologically, the episode name is based on Schoenfeld’s ( 1985a , 1985b ) analysis, because we observed that, similar to problem-solving, posers identify what possibilities for problem posing the given situations provide through their conditions.

Pelczer and Gamboa ( 2009 ) have aspects of variation in the stage of formulation in which a problem is written down and the formulation is evaluated. Problem formulating can also be found in the model by Cruz ( 2006 ). The principle of variation also plays a central role in problem solving. Schoenfeld ( 1985b ), for example, suggests posing modified problems by replacing or varying the conditions of a particular problem that is difficult to solve.

Koichu and Kontorovich ( 2013 ) consider spontaneously associated problems related to a given problem-posing situation in their model, yet this is only one aspect of the generation described above. The distinction between variation and generation is theoretically already conceptualized by Silver ( 1994 ). In empirical studies on problem posing, there are so far no objective criteria that enable distinct identification of both activities. The phase model at hand proposes criteria for this distinction.

Problem-solving

Cruz ( 2006 ) explicitly mentions problem solving as a stage in his problem-posing phase model. In the model by Pelczer and Gamboa ( 2009 ), problem solving is implicit in the evaluation phase, in which the posed problem is assessed and modified. This is presumably done based on the solution of it.

The stage of evaluation in the phase model by Pelczer and Gamboa ( 2009 ) shares the same name and has similar characteristics. Cruz ( 2006 ) implicitly considers evaluation when the posers improve the posed problem when they deem it not suitable for a specific learning group. The activity of evaluation is closely related to the metacognitive activity of the regulation of cognition (Flavell, 1979 ; Schraw & Moshman, 1995 ). In research on problem posing, there are hardly any studies that investigate metacognitive behavior, yet some frameworks implicitly include aspects of it. Kontorovich et al. ( 2012 ), for example, consider aptness by means of fitness, suitableness, and appropriateness of a posed problem.

4.2 Derivation of a descriptive phase model for problem posing

There is no predetermined order of episode types which means there can be transitions from any episode type to any other. However, there is a kind of “natural order” in which episode types appear in most processes and in which transitions often occur. This has been indicated by the order in which the episode types were presented in Sect. 4.1. It was observed that first the conditions of a situation are grasped ( situation analysis ) and then new tasks are posed through variation or generation ; these tasks are solved in order to evaluate them based on the solution. Of course, we did not observe exactly this order in every process, but across the participants and the different problem-posing situations, parts of this superordinate pattern were identified. Often the situation analysis was observed at the beginning of the process and at the end of a longer phase of variation . Also typical were frequent changes between variation or generation and problem solving (sometimes in combination with evaluation ). Furthermore, problem posing was identified as a cyclical activity. Several participants were observed to revise or to further vary their previously posed problems. Figure 2 shows the T&U’S process following Schoenfeld’s ( 1985b ) illustrations of problem-solving processes. Several characteristic transitions can be observed in this process. The vertical lines shown in this figure indicate points in time when a new task (either by variation or by generation) was posed.

Example of a timeline chart of the problem-posing process by Theresa and Ugur as described in Sect. 4 following the illustrations by Schoenfeld ( 1985b )

From these theoretically justifiable as well as empirically observable patterns in the sequence of episodes, the descriptive phase model shown in Fig. 3 was derived. It contains all five content-related episodes as a complete graph. All transitions indicated by arrows can occur and have been observed empirically in the study. However, not all episode types need to occur in a process. Several participants were observed to revise or to further vary their previously posed problems. In addition, in most cases, not only one but several problems are posed in numerous cycles. The model reflects this observation through its cyclic structure. The model is used to represent all these possible paths within the problem-posing process.

Descriptive phase model for problem posing based on structured situations

To check the interrater agreement, 30.7% of the total video material of 7 h and 46 min was coded by a second independent rater and combined into an agreement table (see Table 3 ) using the EasyDIAg algorithm (Holle & Rein, 2015 ). As explained in Sect. 4.1.5, the episode types of problem solving and evaluation have empirically often been observed simultaneously, which is why simultaneous coding was allowed. We have, therefore, considered this simultaneous coding as a separate category for the verification of interrater agreement. If the start or end of a process was coded differently in time by the two raters, there are unlinked events in the agreement which are coded as X. The entry X–X in Table 3 can, therefore, not occur empirically.

With a Cohen’s kappa of κ = 0.81, the interrater agreement is almost perfect (Landis & Koch, 1977 , p. 165). This high level of agreement is particularly gratifying as the evaluation method is a highly subjective and interpretative procedure, yet the developed categories are capable of consistent coding. As anticipated, the biggest coding differences are observed for the categories variation and generation as well as the distinction between the categories of problem solving, problem solving and evaluation, and evaluation . The kappa calculated for the separate categories are (with the abbreviations from Table 3 as indices) κ SA = 0.87, κ V = 0.8 3, κ G = 0.7 2, κ PS = 0.8 7, κ PS/E = 0.7 3, κ E = 0.4 9, and κ O = 97.

5 Discussion

This study aimed to develop a valid and reliable model to describe and analyze problem-posing processes. Schoenfeld ( 2000 ) provides eight criteria for evaluating models in mathematics education: (i) descriptive power, (ii) explanatory power, (iii) scope, (iv) predictive power, (v) rigor and specificity, (vi) falsifiability, (vii) replicability, and (viii) multiple sources of evidence. Criteria (i), (iii), (v), and (vii) will be outlined to discuss the potential and limitations of the presented framework.

Regarding research question (1), five content-related episode types— situation analysis , variation , generation , problem solving , and evaluation— were identified inductively which enable objective coding through their operationalization. The episode types of the developed phase model enable a specific descriptive perspective on all observed problem-posing processes in the study in a time-covering manner. This description, we argue, provides a better understanding of problem-posing processes in general (i). Furthermore and with regard to research question (2), from the observed processes, a general structure in terms of the sequence of the episodes was identified from which we were able to derive a descriptive process model for problem posing. The high interrater agreement attests to the replicability of the model (vii). The participants of the study were heterogeneous and ranged from PSTs in the first bachelor’s semester for primary school to PSTs in the 3rd master’s semester for high school. Equally heterogeneous were the processes that could nevertheless be analyzed by the developed model (iii). The detailed descriptions, coding instructions, and theoretical classifications provide specificity to the terms. In the online supplement , anchor examples serve for additional specification (v).

The model developed here provides additional insights compared to existing models (e.g., Cruz, 2006 ; Pelczer & Gamboa, 2009 ): It distinguishes the episode types variation and generation empirically which Silver ( 1994 ) already conceptualized theoretically. Additionally, the model encompasses non-content-related episodes for the description that have also been identified in descriptive models of problem-solving (Rott et al., 2021 ).

The phase model can now be used to characterize, for example, different degrees of quality of the problem-posing process which is still a recent topic in problem-posing research and for which considering the products and processes seems advisable (Kontorovich & Koichu, 2016 ; Patáková, 2014 ; Rosli et al., 2013 ; Singer & Voica, 2017 ). Thus, as in problem-solving research (cf. Schoenfeld, 1985b ), a comparison between experts and novices might be a fruitful approach to identify different types of problem posers. Furthermore and following the process-oriented research on problem-solving (Rott et al., 2021 ), it would be conceivable that the process of posing routine tasks proceeds differently than the process of posing non-routine problems.

Finally, possible limitations to the generalizability of the developed model will be addressed. In general, the model offers one possible perspective on problem-posing processes. Depending on the selected problem-posing situation, sample, or study design, it cannot be ruled out that slightly different or even additional episode types may also occur. We also find other perspectives on problem-posing processes in research (e.g., Headrick et al., 2020 ). This study considers two specific structured situations with a non-routine initial problem. However, the developed phase model has also been successfully applied to situations with routine initial problems and other mathematical contents within bachelor and master theses. With small changes, the model was also successfully applied to processes based on unstructured situations in several master theses. Moreover, this study has PSTs as a sample. The phase model was successfully applied in bachelor and master theses to other sample groups such as school students and teachers (iii). Therefore, there are strong indications that support the generalizability of the phase model, which could still be clarified in follow-up studies.

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Baumanns, L., Rott, B. The process of problem posing: development of a descriptive phase model of problem posing. Educ Stud Math 110 , 251–269 (2022). https://doi.org/10.1007/s10649-021-10136-y

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