Identify Goal
Define Problem
Define Problem
Gather Data
Define Causes
Identify Options
Clarify Problem
Generate Ideas
Evaluate Options
Generate Ideas
Choose the Best Solution
Implement Solution
Select Solution
Take Action
MacLeod offers her own problem solving procedure, which echoes the above steps:
“1. Recognize the Problem: State what you see. Sometimes the problem is covert. 2. Identify: Get the facts — What exactly happened? What is the issue? 3. and 4. Explore and Connect: Dig deeper and encourage group members to relate their similar experiences. Now you're getting more into the feelings and background [of the situation], not just the facts. 5. Possible Solutions: Consider and brainstorm ideas for resolution. 6. Implement: Choose a solution and try it out — this could be role play and/or a discussion of how the solution would be put in place. 7. Evaluate: Revisit to see if the solution was successful or not.”
Many of these problem solving techniques can be used in concert with one another, or multiple can be appropriate for any given problem. It’s less about facilitating a perfect CPS session, and more about encouraging team members to continually think outside the box and push beyond personal boundaries that inhibit their innovative thinking. So, try out several methods, find those that resonate best with your team, and continue adopting new techniques and adapting your processes along the way.
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All teams and organizations encounter challenges. Approaching those challenges without a structured problem solving process can end up making things worse.
Proven problem solving techniques such as those outlined below can guide your group through a process of identifying problems and challenges , ideating on possible solutions , and then evaluating and implementing the most suitable .
In this post, you'll find problem-solving tools you can use to develop effective solutions. You'll also find some tips for facilitating the problem solving process and solving complex problems.
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Problem solving is a process of finding and implementing a solution to a challenge or obstacle. In most contexts, this means going through a problem solving process that begins with identifying the issue, exploring its root causes, ideating and refining possible solutions before implementing and measuring the impact of that solution.
For simple or small problems, it can be tempting to skip straight to implementing what you believe is the right solution. The danger with this approach is that without exploring the true causes of the issue, it might just occur again or your chosen solution may cause other issues.
Particularly in the world of work, good problem solving means using data to back up each step of the process, bringing in new perspectives and effectively measuring the impact of your solution.
Effective problem solving can help ensure that your team or organization is well positioned to overcome challenges, be resilient to change and create innovation. In my experience, problem solving is a combination of skillset, mindset and process, and it’s especially vital for leaders to cultivate this skill.
A problem solving process is a step-by-step framework from going from discovering a problem all the way through to implementing a solution.
With practice, this framework can become intuitive, and innovative companies tend to have a consistent and ongoing ability to discover and tackle challenges when they come up.
You might see everything from a four step problem solving process through to seven steps. While all these processes cover roughly the same ground, I’ve found a seven step problem solving process is helpful for making all key steps legible.
We’ll outline that process here and then follow with techniques you can use to explore and work on that step of the problem solving process with a group.
The seven-step problem solving process is:
The first stage of any problem solving process is to identify the problem(s) you need to solve. This often looks like using group discussions and activities to help a group surface and effectively articulate the challenges they’re facing and wish to resolve.
Be sure to align with your team on the exact definition and nature of the problem you’re solving. An effective process is one where everyone is pulling in the same direction – ensure clarity and alignment now to help avoid misunderstandings later.
The process of problem analysis means ensuring that the problem you are seeking to solve is the right problem . Choosing the right problem to solve means you are on the right path to creating the right solution.
At this stage, you may look deeper at the problem you identified to try and discover the root cause at the level of people or process. You may also spend some time sourcing data, consulting relevant parties and creating and refining a problem statement.
Problem refinement means adjusting scope or focus of the problem you will be aiming to solve based on what comes up during your analysis. As you analyze data sources, you might discover that the root cause means you need to adjust your problem statement. Alternatively, you might find that your original problem statement is too big to be meaningful approached within your current project.
Remember that the goal of any problem refinement is to help set the stage for effective solution development and deployment. Set the right focus and get buy-in from your team here and you’ll be well positioned to move forward with confidence.
Once your group has nailed down the particulars of the problem you wish to solve, you want to encourage a free flow of ideas connecting to solving that problem. This can take the form of problem solving games that encourage creative thinking or techniquess designed to produce working prototypes of possible solutions.
The key to ensuring the success of this stage of the problem solving process is to encourage quick, creative thinking and create an open space where all ideas are considered. The best solutions can often come from unlikely places and by using problem solving techniques that celebrate invention, you might come up with solution gold.
No solution is perfect right out of the gate. It’s important to discuss and develop the solutions your group has come up with over the course of following the previous problem solving steps in order to arrive at the best possible solution. Problem solving games used in this stage involve lots of critical thinking, measuring potential effort and impact, and looking at possible solutions analytically.
During this stage, you will often ask your team to iterate and improve upon your front-running solutions and develop them further. Remember that problem solving strategies always benefit from a multitude of voices and opinions, and not to let ego get involved when it comes to choosing which solutions to develop and take further.
Finding the best solution is the goal of all problem solving workshops and here is the place to ensure that your solution is well thought out, sufficiently robust and fit for purpose.
Nearly there! Once you’ve got a set of possible, you’ll need to make a decision on which to implement. This can be a consensus-based group decision or it might be for a leader or major stakeholder to decide. You’ll find a set of effective decision making methods below.
Once your group has reached consensus and selected a solution, there are some additional actions that also need to be decided upon. You’ll want to work on allocating ownership of the project, figure out who will do what, how the success of the solution will be measured and decide the next course of action.
Set clear accountabilities, actions, timeframes, and follow-ups for your chosen solution. Make these decisions and set clear next-steps in the problem solving workshop so that everyone is aligned and you can move forward effectively as a group.
Ensuring that you plan for the roll-out of a solution is one of the most important problem solving steps. Without adequate planning or oversight, it can prove impossible to measure success or iterate further if the problem was not solved.
This is what we were waiting for! All problem solving processes have the end goal of implementing an effective and impactful solution that your group has confidence in.
Project management and communication skills are key here – your solution may need to adjust when out in the wild or you might discover new challenges along the way. For some solutions, you might also implement a test with a small group and monitor results before rolling it out to an entire company.
You should have a clear owner for your solution who will oversee the plans you made together and help ensure they’re put into place. This person will often coordinate the implementation team and set-up processes to measure the efficacy of your solution too.
So you and your team developed a great solution to a problem and have a gut feeling it’s been solved. Work done, right? Wrong. All problem solving strategies benefit from evaluation, consideration, and feedback.
You might find that the solution does not work for everyone, might create new problems, or is potentially so successful that you will want to roll it out to larger teams or as part of other initiatives.
None of that is possible without taking the time to evaluate the success of the solution you developed in your problem solving model and adjust if necessary.
Remember that the problem solving process is often iterative and it can be common to not solve complex issues on the first try. Even when this is the case, you and your team will have generated learning that will be important for future problem solving workshops or in other parts of the organization.
It’s also worth underlining how important record keeping is throughout the problem solving process. If a solution didn’t work, you need to have the data and records to see why that was the case. If you go back to the drawing board, notes from the previous workshop can help save time.
Every effective problem solving process begins with an agenda . In our experience, a well-structured problem solving workshop is one of the best methods for successfully guiding a group from exploring a problem to implementing a solution.
The format of a workshop ensures that you can get buy-in from your group, encourage free-thinking and solution exploration before making a decision on what to implement following the session.
This Design Sprint 2.0 template is an effective problem solving process from top agency AJ&Smart. It’s a great format for the entire problem solving process, with four-days of workshops designed to surface issues, explore solutions and even test a solution.
Check it for an example of how you might structure and run a problem solving process and feel free to copy and adjust it your needs!
For a shorter process you can run in a single afternoon, this remote problem solving agenda will guide you effectively in just a couple of hours.
Whatever the length of your workshop, by using SessionLab, it’s easy to go from an idea to a complete agenda . Start by dragging and dropping your core problem solving activities into place . Add timings, breaks and necessary materials before sharing your agenda with your colleagues.
The resulting agenda will be your guide to an effective and productive problem solving session that will also help you stay organized on the day!
In this section, we’ll look at in-depth problem-solving methods that provide a complete end-to-end process for developing effective solutions. These will help guide your team from the discovery and definition of a problem through to delivering the right solution.
If you’re looking for an all-encompassing method or problem-solving model, these processes are a great place to start. They’ll ask your team to challenge preconceived ideas and adopt a mindset for solving problems more effectively.
Individual approaches to solving a problem can be very different based on what team or role an individual holds. It can be easy for existing biases or perspectives to find their way into the mix, or for internal politics to direct a conversation.
Six Thinking Hats is a classic method for identifying the problems that need to be solved and enables your team to consider them from different angles, whether that is by focusing on facts and data, creative solutions, or by considering why a particular solution might not work.
Like all problem-solving frameworks, Six Thinking Hats is effective at helping teams remove roadblocks from a conversation or discussion and come to terms with all the aspects necessary to solve complex problems.
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.
Featured courtesy of Jonathan Courtney of AJ&Smart Berlin, Lightning Decision Jam is one of those strategies that should be in every facilitation toolbox. Exploring problems and finding solutions is often creative in nature, though as with any creative process, there is the potential to lose focus and get lost.
Unstructured discussions might get you there in the end, but it’s much more effective to use a method that creates a clear process and team focus.
In Lightning Decision Jam, participants are invited to begin by writing challenges, concerns, or mistakes on post-its without discussing them before then being invited by the moderator to present them to the group.
From there, the team vote on which problems to solve and are guided through steps that will allow them to reframe those problems, create solutions and then decide what to execute on.
By deciding the problems that need to be solved as a team before moving on, this group process is great for ensuring the whole team is aligned and can take ownership over the next stages.
Lightning Decision Jam (LDJ) #action #decision making #problem solving #issue analysis #innovation #design #remote-friendly It doesn’t matter where you work and what your job role is, if you work with other people together as a team, you will always encounter the same challenges: Unclear goals and miscommunication that cause busy work and overtime Unstructured meetings that leave attendants tired, confused and without clear outcomes. Frustration builds up because internal challenges to productivity are not addressed Sudden changes in priorities lead to a loss of focus and momentum Muddled compromise takes the place of clear decision- making, leaving everybody to come up with their own interpretation. In short, a lack of structure leads to a waste of time and effort, projects that drag on for too long and frustrated, burnt out teams. AJ&Smart has worked with some of the most innovative, productive companies in the world. What sets their teams apart from others is not better tools, bigger talent or more beautiful offices. The secret sauce to becoming a more productive, more creative and happier team is simple: Replace all open discussion or brainstorming with a structured process that leads to more ideas, clearer decisions and better outcomes. When a good process provides guardrails and a clear path to follow, it becomes easier to come up with ideas, make decisions and solve problems. This is why AJ&Smart created Lightning Decision Jam (LDJ). It’s a simple and short, but powerful group exercise that can be run either in-person, in the same room, or remotely with distributed teams.
While problems can be complex, the problem-solving methods you use to identify and solve those problems can often be simple in design.
By taking the time to truly identify and define a problem before asking the group to reframe the challenge as an opportunity, this method is a great way to enable change.
Begin by identifying a focus question and exploring the ways in which it manifests before splitting into five teams who will each consider the problem using a different method: escape, reversal, exaggeration, distortion or wishful. Teams develop a problem objective and create ideas in line with their method before then feeding them back to the group.
This method is great for enabling in-depth discussions while also creating space for finding creative solutions too!
Problem Definition #problem solving #idea generation #creativity #online #remote-friendly A problem solving technique to define a problem, challenge or opportunity and to generate ideas.
Sometimes, a group needs to go further with their strategies and analyze the root cause at the heart of organizational issues. An RCA or root cause analysis is the process of identifying what is at the heart of business problems or recurring challenges.
The 5 Whys is a simple and effective method of helping a group go find the root cause of any problem or challenge and conduct analysis that will deliver results.
By beginning with the creation of a problem statement and going through five stages to refine it, The 5 Whys provides everything you need to truly discover the cause of an issue.
The 5 Whys #hyperisland #innovation This simple and powerful method is useful for getting to the core of a problem or challenge. As the title suggests, the group defines a problems, then asks the question “why” five times, often using the resulting explanation as a starting point for creative problem solving.
World Cafe is a simple but powerful facilitation technique to help bigger groups to focus their energy and attention on solving complex problems.
World Cafe enables this approach by creating a relaxed atmosphere where participants are able to self-organize and explore topics relevant and important to them which are themed around a central problem-solving purpose. Create the right atmosphere by modeling your space after a cafe and after guiding the group through the method, let them take the lead!
Making problem-solving a part of your organization’s culture in the long term can be a difficult undertaking. More approachable formats like World Cafe can be especially effective in bringing people unfamiliar with workshops into the fold.
World Cafe #hyperisland #innovation #issue analysis World Café is a simple yet powerful method, originated by Juanita Brown, for enabling meaningful conversations driven completely by participants and the topics that are relevant and important to them. Facilitators create a cafe-style space and provide simple guidelines. Participants then self-organize and explore a set of relevant topics or questions for conversation.
One of the best approaches is to create a safe space for a group to share and discover practices and behaviors that can help them find their own solutions.
With DAD, you can help a group choose which problems they wish to solve and which approaches they will take to do so. It’s great at helping remove resistance to change and can help get buy-in at every level too!
This process of enabling frontline ownership is great in ensuring follow-through and is one of the methods you will want in your toolbox as a facilitator.
Discovery & Action Dialogue (DAD) #idea generation #liberating structures #action #issue analysis #remote-friendly DADs make it easy for a group or community to discover practices and behaviors that enable some individuals (without access to special resources and facing the same constraints) to find better solutions than their peers to common problems. These are called positive deviant (PD) behaviors and practices. DADs make it possible for people in the group, unit, or community to discover by themselves these PD practices. DADs also create favorable conditions for stimulating participants’ creativity in spaces where they can feel safe to invent new and more effective practices. Resistance to change evaporates as participants are unleashed to choose freely which practices they will adopt or try and which problems they will tackle. DADs make it possible to achieve frontline ownership of solutions.
Design Sprint 2.0
Want to see how a team can solve big problems and move forward with prototyping and testing solutions in a few days? The Design Sprint 2.0 template from Jake Knapp, author of Sprint, is a complete agenda for a with proven results.
Developing the right agenda can involve difficult but necessary planning. Ensuring all the correct steps are followed can also be stressful or time-consuming depending on your level of experience.
Use this complete 4-day workshop template if you are finding there is no obvious solution to your challenge and want to focus your team around a specific problem that might require a shortcut to launching a minimum viable product or waiting for the organization-wide implementation of a solution.
Open space technology- developed by Harrison Owen – creates a space where large groups are invited to take ownership of their problem solving and lead individual sessions. Open space technology is a great format when you have a great deal of expertise and insight in the room and want to allow for different takes and approaches on a particular theme or problem you need to be solved.
Start by bringing your participants together to align around a central theme and focus their efforts. Explain the ground rules to help guide the problem-solving process and then invite members to identify any issue connecting to the central theme that they are interested in and are prepared to take responsibility for.
Once participants have decided on their approach to the core theme, they write their issue on a piece of paper, announce it to the group, pick a session time and place, and post the paper on the wall. As the wall fills up with sessions, the group is then invited to join the sessions that interest them the most and which they can contribute to, then you’re ready to begin!
Everyone joins the problem-solving group they’ve signed up to, record the discussion and if appropriate, findings can then be shared with the rest of the group afterward.
Open Space Technology #action plan #idea generation #problem solving #issue analysis #large group #online #remote-friendly Open Space is a methodology for large groups to create their agenda discerning important topics for discussion, suitable for conferences, community gatherings and whole system facilitation
Using a problem-solving method to help a team identify and analyze a problem can be a quick and effective addition to any workshop or meeting.
While further actions are always necessary, you can generate momentum and alignment easily, and these activities are a great place to get started.
We’ve put together this list of techniques to help you and your team with problem identification, analysis, and discussion that sets the foundation for developing effective solutions.
Let’s take a look!
Organizational or team challenges are rarely simple, and it’s important to remember that one problem can be an indication of something that goes deeper and may require further consideration to be solved.
Fishbone Analysis helps groups to dig deeper and understand the origins of a problem. It’s a great example of a root cause analysis method that is simple for everyone on a team to get their head around.
Participants in this activity are asked to annotate a diagram of a fish, first adding the problem or issue to be worked on at the head of a fish before then brainstorming the root causes of the problem and adding them as bones on the fish.
Using abstractions such as a diagram of a fish can really help a team break out of their regular thinking and develop a creative approach.
Fishbone Analysis #problem solving ##root cause analysis #decision making #online facilitation A process to help identify and understand the origins of problems, issues or observations.
Encouraging visual thinking can be an essential part of many strategies. By simply reframing and clarifying problems, a group can move towards developing a problem solving model that works for them.
In Problem Tree, groups are asked to first brainstorm a list of problems – these can be design problems, team problems or larger business problems – and then organize them into a hierarchy. The hierarchy could be from most important to least important or abstract to practical, though the key thing with problem solving games that involve this aspect is that your group has some way of managing and sorting all the issues that are raised.
Once you have a list of problems that need to be solved and have organized them accordingly, you’re then well-positioned for the next problem solving steps.
Problem tree #define intentions #create #design #issue analysis A problem tree is a tool to clarify the hierarchy of problems addressed by the team within a design project; it represents high level problems or related sublevel problems.
Chances are you’ve heard of the SWOT Analysis before. This problem-solving method focuses on identifying strengths, weaknesses, opportunities, and threats is a tried and tested method for both individuals and teams.
Start by creating a desired end state or outcome and bare this in mind – any process solving model is made more effective by knowing what you are moving towards. Create a quadrant made up of the four categories of a SWOT analysis and ask participants to generate ideas based on each of those quadrants.
Once you have those ideas assembled in their quadrants, cluster them together based on their affinity with other ideas. These clusters are then used to facilitate group conversations and move things forward.
SWOT analysis #gamestorming #problem solving #action #meeting facilitation The SWOT Analysis is a long-standing technique of looking at what we have, with respect to the desired end state, as well as what we could improve on. It gives us an opportunity to gauge approaching opportunities and dangers, and assess the seriousness of the conditions that affect our future. When we understand those conditions, we can influence what comes next.
Not every problem-solving approach is right for every challenge, and deciding on the right method for the challenge at hand is a key part of being an effective team.
The Agreement Certainty matrix helps teams align on the nature of the challenges facing them. By sorting problems from simple to chaotic, your team can understand what methods are suitable for each problem and what they can do to ensure effective results.
If you are already using Liberating Structures techniques as part of your problem-solving strategy, the Agreement-Certainty Matrix can be an invaluable addition to your process. We’ve found it particularly if you are having issues with recurring problems in your organization and want to go deeper in understanding the root cause.
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.
Organizing and charting a team’s progress can be important in ensuring its success. SQUID (Sequential Question and Insight Diagram) is a great model that allows a team to effectively switch between giving questions and answers and develop the skills they need to stay on track throughout the process.
Begin with two different colored sticky notes – one for questions and one for answers – and with your central topic (the head of the squid) on the board. Ask the group to first come up with a series of questions connected to their best guess of how to approach the topic. Ask the group to come up with answers to those questions, fix them to the board and connect them with a line. After some discussion, go back to question mode by responding to the generated answers or other points on the board.
It’s rewarding to see a diagram grow throughout the exercise, and a completed SQUID can provide a visual resource for future effort and as an example for other teams.
SQUID #gamestorming #project planning #issue analysis #problem solving When exploring an information space, it’s important for a group to know where they are at any given time. By using SQUID, a group charts out the territory as they go and can navigate accordingly. SQUID stands for Sequential Question and Insight Diagram.
To continue with our nautical theme, Speed Boat is a short and sweet activity that can help a team quickly identify what employees, clients or service users might have a problem with and analyze what might be standing in the way of achieving a solution.
Methods that allow for a group to make observations, have insights and obtain those eureka moments quickly are invaluable when trying to solve complex problems.
In Speed Boat, the approach is to first consider what anchors and challenges might be holding an organization (or boat) back. Bonus points if you are able to identify any sharks in the water and develop ideas that can also deal with competitors!
Speed Boat #gamestorming #problem solving #action Speedboat is a short and sweet way to identify what your employees or clients don’t like about your product/service or what’s standing in the way of a desired goal.
Some of the most effective ways of solving problems is by encouraging teams to be more inclusive and diverse in their thinking.
Based on the six key questions journalism students are taught to answer in articles and news stories, The Journalistic Six helps create teams to see the whole picture. By using who, what, when, where, why, and how to facilitate the conversation and encourage creative thinking, your team can make sure that the problem identification and problem analysis stages of the are covered exhaustively and thoughtfully. Reporter’s notebook and dictaphone optional.
The Journalistic Six – Who What When Where Why How #idea generation #issue analysis #problem solving #online #creative thinking #remote-friendly A questioning method for generating, explaining, investigating ideas.
Individual and group perspectives are incredibly important, but what happens if people are set in their minds and need a change of perspective in order to approach a problem more effectively?
Flip It is a method we love because it is both simple to understand and run, and allows groups to understand how their perspectives and biases are formed.
Participants in Flip It are first invited to consider concerns, issues, or problems from a perspective of fear and write them on a flip chart. Then, the group is asked to consider those same issues from a perspective of hope and flip their understanding.
No problem and solution is free from existing bias and by changing perspectives with Flip It, you can then develop a problem solving model quickly and effectively.
Flip It! #gamestorming #problem solving #action Often, a change in a problem or situation comes simply from a change in our perspectives. Flip It! is a quick game designed to show players that perspectives are made, not born.
Now for an activity that is a little out of the (toy) box. LEGO Serious Play is a facilitation methodology that can be used to improve creative thinking and problem-solving skills.
The LEGO Challenge includes giving each member of the team an assignment that is hidden from the rest of the group while they create a structure without speaking.
What the LEGO challenge brings to the table is a fun working example of working with stakeholders who might not be on the same page to solve problems. Also, it’s LEGO! Who doesn’t love LEGO!
LEGO Challenge #hyperisland #team A team-building activity in which groups must work together to build a structure out of LEGO, but each individual has a secret “assignment” which makes the collaborative process more challenging. It emphasizes group communication, leadership dynamics, conflict, cooperation, patience and problem solving strategy.
If not carefully managed, the problem identification and problem analysis stages of the problem-solving process can actually create more problems and misunderstandings.
The What, So What, Now What? problem-solving activity is designed to help collect insights and move forward while also eliminating the possibility of disagreement when it comes to identifying, clarifying, and analyzing organizational or work problems.
Facilitation is all about bringing groups together so that might work on a shared goal and the best problem-solving strategies ensure that teams are aligned in purpose, if not initially in opinion or insight.
Throughout the three steps of this game, you give everyone on a team to reflect on a problem by asking what happened, why it is important, and what actions should then be taken.
This can be a great activity for bringing our individual perceptions about a problem or challenge and contextualizing it in a larger group setting. This is one of the most important problem-solving skills you can bring to your organization.
W³ – What, So What, Now What? #issue analysis #innovation #liberating structures You can help groups reflect on a shared experience in a way that builds understanding and spurs coordinated action while avoiding unproductive conflict. It is possible for every voice to be heard while simultaneously sifting for insights and shaping new direction. Progressing in stages makes this practical—from collecting facts about What Happened to making sense of these facts with So What and finally to what actions logically follow with Now What . The shared progression eliminates most of the misunderstandings that otherwise fuel disagreements about what to do. Voila!
Problem analysis can be one of the most important and decisive stages of all problem-solving tools. Sometimes, a team can become bogged down in the details and are unable to move forward.
Journalists is an activity that can avoid a group from getting stuck in the problem identification or problem analysis stages of the process.
In Journalists, the group is invited to draft the front page of a fictional newspaper and figure out what stories deserve to be on the cover and what headlines those stories will have. By reframing how your problems and challenges are approached, you can help a team move productively through the process and be better prepared for the steps to follow.
Journalists #vision #big picture #issue analysis #remote-friendly This is an exercise to use when the group gets stuck in details and struggles to see the big picture. Also good for defining a vision.
Now you have the context and background of the problem you are trying to solving, now comes the time to start ideating and thinking about how you’ll solve the issue.
Here, you’ll want to encourage creative, free thinking and speed. Get as many ideas out as possible and explore different perspectives so you have the raw material for the next step.
Looking at a problem from a new angle can be one of the most effective ways of creating an effective solution. TRIZ is a problem-solving tool that asks the group to consider what they must not do in order to solve a challenge.
By reversing the discussion, new topics and taboo subjects often emerge, allowing the group to think more deeply and create ideas that confront the status quo in a safe and meaningful way. If you’re working on a problem that you’ve tried to solve before, TRIZ is a great problem-solving method to help your team get unblocked.
Making Space with TRIZ #issue analysis #liberating structures #issue resolution You can clear space for innovation by helping a group let go of what it knows (but rarely admits) limits its success and by inviting creative destruction. TRIZ makes it possible to challenge sacred cows safely and encourages heretical thinking. The question “What must we stop doing to make progress on our deepest purpose?” induces seriously fun yet very courageous conversations. Since laughter often erupts, issues that are otherwise taboo get a chance to be aired and confronted. With creative destruction come opportunities for renewal as local action and innovation rush in to fill the vacuum. Whoosh!
Brainstorming is part of the bread and butter of the problem-solving process and all problem-solving strategies benefit from getting ideas out and challenging a team to generate solutions quickly.
With Mindspin, participants are encouraged not only to generate ideas but to do so under time constraints and by slamming down cards and passing them on. By doing multiple rounds, your team can begin with a free generation of possible solutions before moving on to developing those solutions and encouraging further ideation.
This is one of our favorite problem-solving activities and can be great for keeping the energy up throughout the workshop. Remember the importance of helping people become engaged in the process – energizing problem-solving techniques like Mindspin can help ensure your team stays engaged and happy, even when the problems they’re coming together to solve are complex.
MindSpin #teampedia #idea generation #problem solving #action A fast and loud method to enhance brainstorming within a team. Since this activity has more than round ideas that are repetitive can be ruled out leaving more creative and innovative answers to the challenge.
One of the most useful problem solving skills you can teach your team is of approaching challenges with creativity, flexibility, and openness. Games like The Creativity Dice allow teams to overcome the potential hurdle of too much linear thinking and approach the process with a sense of fun and speed.
In The Creativity Dice, participants are organized around a topic and roll a dice to determine what they will work on for a period of 3 minutes at a time. They might roll a 3 and work on investigating factual information on the chosen topic. They might roll a 1 and work on identifying the specific goals, standards, or criteria for the session.
Encouraging rapid work and iteration while asking participants to be flexible are great skills to cultivate. Having a stage for idea incubation in this game is also important. Moments of pause can help ensure the ideas that are put forward are the most suitable.
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.
Brainstorming without structure can quickly become chaotic or frustrating. In a problem-solving context, having an ideation framework to follow can help ensure your team is both creative and disciplined.
In this method, you’ll find an idea generation process that encourages your group to brainstorm effectively before developing their ideas and begin clustering them together. By using concepts such as Yes and…, more is more and postponing judgement, you can create the ideal conditions for brainstorming with ease.
Idea & Concept Development #hyperisland #innovation #idea generation Ideation and Concept Development is a process for groups to work creatively and collaboratively to generate creative ideas. It’s a general approach that can be adapted and customized to suit many different scenarios. It includes basic principles for idea generation and several steps for groups to work with. It also includes steps for idea selection and development.
The success of any problem-solving process can be measured by the solutions it produces. After you’ve defined the issue, explored existing ideas, and ideated, it’s time to develop and refine your ideas in order to bring them closer to a solution that actually solves the problem.
Use these problem-solving techniques when you want to help your team think through their ideas and refine them as part of your problem solving process.
After a team has successfully identified a problem and come up with a few solutions, it can be tempting to call the work of the problem-solving process complete. That said, the first solution is not necessarily the best, and by including a further review and reflection activity into your problem-solving model, you can ensure your group reaches the best possible result.
One of a number of problem-solving games from Thiagi Group, Improved Solutions helps you go the extra mile and develop suggested solutions with close consideration and peer review. By supporting the discussion of several problems at once and by shifting team roles throughout, this problem-solving technique is a dynamic way of finding the best solution.
Improved Solutions #creativity #thiagi #problem solving #action #team You can improve any solution by objectively reviewing its strengths and weaknesses and making suitable adjustments. In this creativity framegame, you improve the solutions to several problems. To maintain objective detachment, you deal with a different problem during each of six rounds and assume different roles (problem owner, consultant, basher, booster, enhancer, and evaluator) during each round. At the conclusion of the activity, each player ends up with two solutions to her problem.
Creative thinking and visual ideation does not need to be confined to the opening stages of your problem-solving strategies. Exercises that include sketching and prototyping on paper can be effective at the solution finding and development stage of the process, and can be great for keeping a team engaged.
By going from simple notes to a crazy 8s round that involves rapidly sketching 8 variations on their ideas before then producing a final solution sketch, the group is able to iterate quickly and visually. Problem-solving techniques like Four-Step Sketch are great if you have a group of different thinkers and want to change things up from a more textual or discussion-based approach.
Four-Step Sketch #design sprint #innovation #idea generation #remote-friendly The four-step sketch is an exercise that helps people to create well-formed concepts through a structured process that includes: Review key information Start design work on paper, Consider multiple variations , Create a detailed solution . This exercise is preceded by a set of other activities allowing the group to clarify the challenge they want to solve. See how the Four Step Sketch exercise fits into a Design Sprint
Ensuring that everyone in a group is able to contribute to a discussion is vital during any problem solving process. Not only does this ensure all bases are covered, but its then easier to get buy-in and accountability when people have been able to contribute to the process.
1-2-4-All is a tried and tested facilitation technique where participants are asked to first brainstorm on a topic on their own. Next, they discuss and share ideas in a pair before moving into a small group. Those groups are then asked to present the best idea from their discussion to the rest of the team.
This method can be used in many different contexts effectively, though I find it particularly shines in the idea development stage of the process. Giving each participant time to concretize their ideas and develop them in progressively larger groups can create a great space for both innovation and psychological safety.
1-2-4-All #idea generation #liberating structures #issue analysis With this facilitation technique you can immediately include everyone regardless of how large the group is. You can generate better ideas and more of them faster than ever before. You can tap the know-how and imagination that is distributed widely in places not known in advance. Open, generative conversation unfolds. Ideas and solutions are sifted in rapid fashion. Most importantly, participants own the ideas, so follow-up and implementation is simplified. No buy-in strategies needed! Simple and elegant!
Some problems are simpler than others and with the right problem-solving activities, you can empower people to take immediate actions that can help create organizational change.
Part of the liberating structures toolkit, 15% solutions is a problem-solving technique that focuses on finding and implementing solutions quickly. A process of iterating and making small changes quickly can help generate momentum and an appetite for solving complex problems.
Problem-solving strategies can live and die on whether people are onboard. Getting some quick wins is a great way of getting people behind the process.
It can be extremely empowering for a team to realize that problem-solving techniques can be deployed quickly and easily and delineate between things they can positively impact and those things they cannot change.
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.
After your group is happy with the possible solutions you’ve developed, now comes the time to choose which to implement. There’s more than one way to make a decision and the best option is often dependant on the needs and set-up of your group.
Sometimes, it’s the case that you’ll want to vote as a group on what is likely to be the most impactful solution. Other times, it might be down to a decision maker or major stakeholder to make the final decision. Whatever your process, here’s some techniques you can use to help you make a decision during your problem solving process.
The problem-solving process is often creative, as complex problems usually require a change of thinking and creative response in order to find the best solutions. While it’s common for the first stages to encourage creative thinking, groups can often gravitate to familiar solutions when it comes to the end of the process.
When selecting solutions, you don’t want to lose your creative energy! The How-Now-Wow Matrix from Gamestorming is a great problem-solving activity that enables a group to stay creative and think out of the box when it comes to selecting the right solution for a given problem.
Problem-solving techniques that encourage creative thinking and the ideation and selection of new solutions can be the most effective in organisational change. Give the How-Now-Wow Matrix a go, and not just for how pleasant it is to say out loud.
How-Now-Wow Matrix #gamestorming #idea generation #remote-friendly When people want to develop new ideas, they most often think out of the box in the brainstorming or divergent phase. However, when it comes to convergence, people often end up picking ideas that are most familiar to them. This is called a ‘creative paradox’ or a ‘creadox’. The How-Now-Wow matrix is an idea selection tool that breaks the creadox by forcing people to weigh each idea on 2 parameters.
All problem-solving techniques hope to not only find solutions to a given problem or challenge but to find the best solution. When it comes to finding a solution, groups are invited to put on their decision-making hats and really think about how a proposed idea would work in practice.
The Impact and Effort Matrix is one of the problem-solving techniques that fall into this camp, empowering participants to first generate ideas and then categorize them into a 2×2 matrix based on impact and effort.
Activities that invite critical thinking while remaining simple are invaluable. Use the Impact and Effort Matrix to move from ideation and towards evaluating potential solutions before then committing to them.
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.
If you’ve followed each of the problem-solving steps with your group successfully, you should move towards the end of your process with heaps of possible solutions developed with a specific problem in mind. But how do you help a group go from ideation to putting a solution into action?
Dotmocracy – or Dot Voting -is a tried and tested method of helping a team in the problem-solving process make decisions and put actions in place with a degree of oversight and consensus.
One of the problem-solving techniques that should be in every facilitator’s toolbox, Dot Voting is fast and effective and can help identify the most popular and best solutions and help bring a group to a decision effectively.
Dotmocracy #action #decision making #group prioritization #hyperisland #remote-friendly Dotmocracy is a simple method for group prioritization or decision-making. It is not an activity on its own, but a method to use in processes where prioritization or decision-making is the aim. The method supports a group to quickly see which options are most popular or relevant. The options or ideas are written on post-its and stuck up on a wall for the whole group to see. Each person votes for the options they think are the strongest, and that information is used to inform a decision.
Straddling the gap between decision making and planning, MoSCoW is a simple and effective method that allows a group team to easily prioritize a set of possible options.
Use this method in a problem solving process by collecting and summarizing all your possible solutions and then categorize them into 4 sections: “Must have”, “Should have”, “Could have”, or “Would like but won‘t get”.
This method is particularly useful when its less about choosing one possible solution and more about prioritorizing which to do first and which may not fit in the scope of your project. In my experience, complex challenges often require multiple small fixes, and this method can be a great way to move from a pile of things you’d all like to do to a structured plan.
MoSCoW #define intentions #create #design #action #remote-friendly MoSCoW is a method that allows the team to prioritize the different features that they will work on. Features are then categorized into “Must have”, “Should have”, “Could have”, or “Would like but won‘t get”. To be used at the beginning of a timeslot (for example during Sprint planning) and when planning is needed.
When it comes to managing the rollout of a solution, clarity and accountability are key factors in ensuring the success of the project. The RAACI chart is a simple but effective model for setting roles and responsibilities as part of a planning session.
Start by listing each person involved in the project and put them into the following groups in order to make it clear who is responsible for what during the rollout of your solution.
Ensure this information is easily accessible and use it to inform who does what and who is looped into discussions and kept up to date.
RAACI #roles and responsibility #teamwork #project management Clarifying roles and responsibilities, levels of autonomy/latitude in decision making, and levels of engagement among diverse stakeholders.
All facilitators know that warm-ups and icebreakers are useful for any workshop or group process. Problem-solving workshops are no different.
Use these problem-solving techniques to warm up a group and prepare them for the rest of the process. Activating your group by tapping into some of the top problem-solving skills can be one of the best ways to see great outcomes from your session.
Solid processes are planned from beginning to end, and the best facilitators know that setting the tone and establishing a safe, open environment can be integral to a successful problem-solving process. Check-in / Check-out is a great way to begin and/or bookend a problem-solving workshop. Checking in to a session emphasizes that everyone will be seen, heard, and expected to contribute.
If you are running a series of meetings, setting a consistent pattern of checking in and checking out can really help your team get into a groove. We recommend this opening-closing activity for small to medium-sized groups though it can work with large groups if they’re disciplined!
Check-in / Check-out #team #opening #closing #hyperisland #remote-friendly Either checking-in or checking-out is a simple way for a team to open or close a process, symbolically and in a collaborative way. Checking-in/out invites each member in a group to be present, seen and heard, and to express a reflection or a feeling. Checking-in emphasizes presence, focus and group commitment; checking-out emphasizes reflection and symbolic closure.
Thinking creatively and not being afraid to make suggestions are important problem-solving skills for any group or team, and warming up by encouraging these behaviors is a great way to start.
Doodling Together is one of our favorite creative ice breaker games – it’s quick, effective, and fun and can make all following problem-solving steps easier by encouraging a group to collaborate visually. By passing cards and adding additional items as they go, the workshop group gets into a groove of co-creation and idea development that is crucial to finding solutions to problems.
Doodling Together #collaboration #creativity #teamwork #fun #team #visual methods #energiser #icebreaker #remote-friendly Create wild, weird and often funny postcards together & establish a group’s creative confidence.
You might remember some version of Show and Tell from being a kid in school and it’s a great problem-solving activity to kick off a session.
Asking participants to prepare a little something before a workshop by bringing an object for show and tell can help them warm up before the session has even begun! Games that include a physical object can also help encourage early engagement before moving onto more big-picture thinking.
By asking your participants to tell stories about why they chose to bring a particular item to the group, you can help teams see things from new perspectives and see both differences and similarities in the way they approach a topic. Great groundwork for approaching a problem-solving process as a team!
Show and Tell #gamestorming #action #opening #meeting facilitation Show and Tell taps into the power of metaphors to reveal players’ underlying assumptions and associations around a topic The aim of the game is to get a deeper understanding of stakeholders’ perspectives on anything—a new project, an organizational restructuring, a shift in the company’s vision or team dynamic.
Who doesn’t love stars? Constellations is a great warm-up activity for any workshop as it gets people up off their feet, energized, and ready to engage in new ways with established topics. It’s also great for showing existing beliefs, biases, and patterns that can come into play as part of your session.
Using warm-up games that help build trust and connection while also allowing for non-verbal responses can be great for easing people into the problem-solving process and encouraging engagement from everyone in the group. Constellations is great in large spaces that allow for movement and is definitely a practical exercise to allow the group to see patterns that are otherwise invisible.
Constellations #trust #connection #opening #coaching #patterns #system Individuals express their response to a statement or idea by standing closer or further from a central object. Used with teams to reveal system, hidden patterns, perspectives.
Problem-solving games that help raise group awareness through a central, unifying metaphor can be effective ways to warm-up a group in any problem-solving model.
Draw a Tree is a simple warm-up activity you can use in any group and which can provide a quick jolt of energy. Start by asking your participants to draw a tree in just 45 seconds – they can choose whether it will be abstract or realistic.
Once the timer is up, ask the group how many people included the roots of the tree and use this as a means to discuss how we can ignore important parts of any system simply because they are not visible.
All problem-solving strategies are made more effective by thinking of problems critically and by exposing things that may not normally come to light. Warm-up games like Draw a Tree are great in that they quickly demonstrate some key problem-solving skills in an accessible and effective way.
Draw a Tree #thiagi #opening #perspectives #remote-friendly With this game you can raise awarness about being more mindful, and aware of the environment we live in.
Each step of the problem-solving workshop benefits from an intelligent deployment of activities, games, and techniques. Bringing your session to an effective close helps ensure that solutions are followed through on and that you also celebrate what has been achieved.
Here are some problem-solving activities you can use to effectively close a workshop or meeting and ensure the great work you’ve done can continue afterward.
Maintaining attention and focus during the closing stages of a problem-solving workshop can be tricky and so being concise when giving feedback can be important. It’s easy to incur “death by feedback” should some team members go on for too long sharing their perspectives in a quick feedback round.
One Breath Feedback is a great closing activity for workshops. You give everyone an opportunity to provide feedback on what they’ve done but only in the space of a single breath. This keeps feedback short and to the point and means that everyone is encouraged to provide the most important piece of feedback to them.
One breath feedback #closing #feedback #action This is a feedback round in just one breath that excels in maintaining attention: each participants is able to speak during just one breath … for most people that’s around 20 to 25 seconds … unless of course you’ve been a deep sea diver in which case you’ll be able to do it for longer.
Matrices feature as part of many effective problem-solving strategies and with good reason. They are easily recognizable, simple to use, and generate results.
The Who What When Matrix is a great tool to use when closing your problem-solving session by attributing a who, what and when to the actions and solutions you have decided upon. The resulting matrix is a simple, easy-to-follow way of ensuring your team can move forward.
Great solutions can’t be enacted without action and ownership. Your problem-solving process should include a stage for allocating tasks to individuals or teams and creating a realistic timeframe for those solutions to be implemented or checked out. Use this method to keep the solution implementation process clear and simple for all involved.
Who/What/When Matrix #gamestorming #action #project planning With Who/What/When matrix, you can connect people with clear actions they have defined and have committed to.
Group discussion can comprise the bulk of most problem-solving activities and by the end of the process, you might find that your team is talked out!
Providing a means for your team to give feedback with short written notes can ensure everyone is head and can contribute without the need to stand up and talk. Depending on the needs of the group, giving an alternative can help ensure everyone can contribute to your problem-solving model in the way that makes the most sense for them.
Response Cards is a great way to close a workshop if you are looking for a gentle warm-down and want to get some swift discussion around some of the feedback that is raised.
Response Cards #debriefing #closing #structured sharing #questions and answers #thiagi #action It can be hard to involve everyone during a closing of a session. Some might stay in the background or get unheard because of louder participants. However, with the use of Response Cards, everyone will be involved in providing feedback or clarify questions at the end of a session.
Problem-solving activities are only one part of the puzzle. While a great method can help unlock your team’s ability to solve problems, without a thoughtful approach and strong facilitation the solutions may not be fit for purpose.
Let’s take a look at some problem-solving tips you can apply to any process to help it be a success!
Jumping straight to solutions can be tempting, though without first clearly articulating a problem, the solution might not be the right one. Many of the problem-solving activities below include sections where the problem is explored and clearly defined before moving on.
This is a vital part of the problem-solving process and taking the time to fully define an issue can save time and effort later. A clear definition helps identify irrelevant information and it also ensures that your team sets off on the right track.
It’s easy for groups to exhibit cognitive bias or have preconceived ideas about both problems and potential solutions. Be sure to back up any problem statements or potential solutions with facts, research, and adequate forethought.
The best techniques ask participants to be methodical and challenge preconceived notions. Make sure you give the group enough time and space to collect relevant information and consider the problem in a new way. By approaching the process with a clear, rational mindset, you’ll often find that better solutions are more forthcoming.
Problems come in all shapes and sizes and so too should the methods you use to solve them. If you find that one approach isn’t yielding results and your team isn’t finding different solutions, try mixing it up. You’ll be surprised at how using a new creative activity can unblock your team and generate great solutions.
Depending on the nature of your team or organizational problems, it’s easy for conversations to get heated. While it’s good for participants to be engaged in the discussions, ensure that emotions don’t run too high and that blame isn’t thrown around while finding solutions.
You’re all in it together, and even if your team or area is seeing problems, that isn’t necessarily a disparagement of you personally. Using facilitation skills to manage group dynamics is one effective method of helping conversations be more constructive.
Your problem-solving method is often only as effective as the group using it. Getting the right people on the job and managing the number of people present is important too!
If the group is too small, you may not get enough different perspectives to effectively solve a problem. If the group is too large, you can go round and round during the ideation stages.
Creating the right group makeup is also important in ensuring you have the necessary expertise and skillset to both identify and follow up on potential solutions. Carefully consider who to include at each stage to help ensure your problem-solving method is followed and positioned for success.
Identifying a problem accurately also requires that all members of a group are able to contribute their views in an open and safe manner.
It can be tough for people to stand up and contribute if the problems or challenges are emotive or personal in nature. Try and create a psychologically safe space for these kinds of discussions and where possible, create regular opportunities for challenges to be brought up organically.
The best solutions can take refinement, iteration, and reflection to come out. Get into a habit of documenting your process in order to keep all the learnings from the session and to allow ideas to mature and develop. Many of the methods below involve the creation of documents or shared resources. Be sure to keep and share these so everyone can benefit from the work done!
Facilitation is all about making group processes easier. With a subject as potentially emotive and important as problem-solving, having an impartial third party in the form of a facilitator can make all the difference in finding great solutions and keeping the process moving. Consider bringing a facilitator to your problem-solving session to get better results and generate meaningful solutions!
It takes time and practice to be an effective problem solver. While some roles or participants might more naturally gravitate towards problem-solving, it can take development and planning to help everyone create better solutions.
You might develop a training program, run a problem-solving workshop or simply ask your team to practice using the techniques below. Check out our post on problem-solving skills to see how you and your group can develop the right mental process and be more resilient to issues too!
Workshops are a great format for solving problems. With the right approach, you can focus a group and help them find the solutions to their own problems. But designing a process can be time-consuming and finding the right activities can be difficult.
Check out our workshop planning guide to level-up your agenda design and start running more effective workshops. Need inspiration? Check out templates designed by expert facilitators to help you kickstart your process!
A structured problem solving process is a surefire way of solving tough problems, discovering creative solutions and driving organizational change. But how can you design for successful outcomes?
With SessionLab, it’s easy to design engaging workshops that deliver results. Drag, drop and reorder blocks to build your agenda. When you make changes or update your agenda, your session timing adjusts automatically , saving you time on manual adjustments.
Collaborating with stakeholders or clients? Share your agenda with a single click and collaborate in real-time. No more sending documents back and forth over email.
Explore how to use SessionLab to design effective problem solving workshops or watch this five minute video to see the planner in action!
The problem-solving process can often be as complicated and multifaceted as the problems they are set-up to solve. With the right problem-solving techniques and a mix of exercises designed to guide discussion and generate purposeful ideas, we hope we’ve given you the tools to find the best solutions as simply and easily as possible.
Is there a problem-solving technique that you are missing here? Do you have a favorite activity or method you use when facilitating? Let us know in the comments below, we’d love to hear from you!
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.
thank you very much for these excellent techniques
Certainly wonderful article, very detailed. Shared!
Your list of techniques for problem solving can be helpfully extended by adding TRIZ to the list of techniques. TRIZ has 40 problem solving techniques derived from methods inventros and patent holders used to get new patents. About 10-12 are general approaches. many organization sponsor classes in TRIZ that are used to solve business problems or general organiztational problems. You can take a look at TRIZ and dwonload a free internet booklet to see if you feel it shound be included per your selection process.
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By Status.net Editorial Team on May 7, 2023 — 5 minutes to read
Definition and importance.
Problem solving is the process of finding solutions to obstacles or challenges you encounter in your life or work. It is a crucial skill that allows you to tackle complex situations, adapt to changes, and overcome difficulties with ease. Mastering this ability will contribute to both your personal and professional growth, leading to more successful outcomes and better decision-making.
The problem-solving process typically includes the following steps:
To start tackling a problem, first, identify and understand it. Analyzing the issue thoroughly helps to clarify its scope and nature. Ask questions to gather information and consider the problem from various angles. Some strategies to define the problem include:
Once the problem is clearly understood, brainstorm possible solutions. Think creatively and keep an open mind, as well as considering lessons from past experiences. Consider:
Evaluate each potential solution, weighing its pros and cons. To facilitate decision-making, use techniques such as:
After evaluating, choose the most suitable solution based on effectiveness, cost, and time constraints.
Implement the chosen solution and monitor its progress. Key actions include:
Utilize feedback from stakeholders and consider potential improvements. Remember that problem-solving is an ongoing process that can always be refined and enhanced.
During each step, you may find it helpful to utilize various problem-solving techniques, such as:
When facing a problem, start by conducting a brainstorming session. Gather your team and encourage an open discussion where everyone contributes ideas, no matter how outlandish they may seem. This helps you:
When brainstorming, remember to:
For effective problem-solving, identifying the root cause of the issue at hand is crucial. Try these methods:
SWOT analysis helps you examine the Strengths, Weaknesses, Opportunities, and Threats related to your problem. To perform a SWOT analysis:
SWOT analysis aids in understanding the internal and external factors affecting the problem, which can help guide your solution.
A mind map is a visual representation of your problem and potential solutions. It enables you to organize information in a structured and intuitive manner. To create a mind map:
Mind mapping allows you to visually see connections between ideas and promotes creativity in problem-solving.
In the business world, you might encounter problems related to finances, operations, or communication. Applying problem-solving skills in these situations could look like:
In educational contexts, problem-solving can be seen in various aspects, such as:
Everyday life is full of challenges that require problem-solving skills. Some examples include:
In this lesson, we’ll discuss mathematical reasoning and methods of problem solving with an eye toward helping your students make the best use of their reasoning skills when it comes to tackling complex problems.
When solving a mathematical problem, it is very common for a student to feel overwhelmed by the information or lack a clear idea about how to get started.
To help the students with their problem-solving “problem,” let’s look at some examples of mathematical problems and some general methods for solving problems:
Identify the following four-digit number when presented with the following information:
Help your students identify and prioritize the information presented.
In this particular example, we want to look for concrete information. Clue #1 tells us that one digit is a 1, but we’re not sure of its location, so we see if we can find a clue with more concrete information.
We can see that clue #5 gives us that kind of information and is the only clue that does, so we start from there.
Because this clue tells us that the thousands place digit is 2, we search for clues relevant to this clue. Clue #2 tells us that the digit in the hundreds place is three times that of the thousands place digit, so it is 6.
So now we need to find the tens and ones place digits, and see that clue #3 tells us that the digit in the ones place is four times the digit in the tens place. But we remember that clue #1 tells us that there’s a one somewhere, and since one is not four times any digit, we see that the one must be in the tens place, which leads us to the conclusion that the digit in the ones place is four. So then we conclude that our number is:
If you were following closely, you would notice that clue #4 was never used. It is a nice way to check our answer, since the digits of 2614 do indeed add up to be thirteen, but we did not need this clue to solve the problem.
Recall that the clues’ relevance were identified and prioritized as follows:
By identifying and prioritizing information, we were able to make the information given in the problem seem less overwhelming. We ordered the clues by relevance, with the most relevant clue providing us with a starting point to solve the problem. This method also utilized the more general method of breaking a problem into smaller and simpler parts to make it easier to solve.
Now let’s look at another mathematical problem and another general problem-solving method to help us solve it:
Two trees with heights of 20 m and 30 m respectively have ropes running from the top of each tree to the bottom of the other tree. The trees are 40 meters apart. We’ll assume that the ropes are pulled tight enough that we can ignore any bending or drooping. How high above the ground do the ropes intersect?
Let’s solve this problem by representing it in a visual way , in this case, a diagram:
You can see that we have a much simpler problem on our hands after drawing the diagram. A, B, C, D, E, and F are vertices of the triangles in the diagram. Now also notice that:
b = the base of triangle EFA
h = the height of triangle EFA and the height above the ground at which the ropes intersect
If we had not drawn this diagram, it would have been very hard to solve this problem, since we need the triangles and their properties to solve for h. Also, this diagram allows us to see that triangle BCA is similar to triangle EFC, and triangle DCA is similar to triangle EFA. Solving for h shows that the ropes intersect twelve meters above the ground.
Students frequently complain that mathematics is too difficult for them, because it is too abstract and unapproachable. Explaining mathematical reasoning and problem solving by using a variety of methods , such as words, numbers, symbols, charts, graphs, tables, diagrams, and concrete models can help students understand the problem better by making it more concrete and approachable.
Let’s try another one.
Given a pickle jar filled with marbles, about how many marbles does the jar contain?
Problems like this one require the student to make and use estimations . In this case, an estimation is all that is required, although, in more complex problems, estimates may help the student arrive at the final answer.
How would a student do this? A good estimation can be found by counting how many marbles are on the base of the jar and multiplying that by the number of marbles that make up the height of the marbles in the jar.
Now to make sure that we understand when and how to use these methods, let’s solve a problem on our own:
How many more faces does a cube have than a square pyramid?
Reveal Answer
The answer is B. To see how many more faces a cube has than a square pyramid, it is best to draw a diagram of a square pyramid and a cube:
From the diagrams above, we can see that the square pyramid has five faces and the cube has six. Therefore, the cube has one more face, so the answer is B.
Before we start having the same problem our model student in the beginning did—that is, being overwhelmed with too much information—let’s have a quick review of all the problem-solving methods we’ve discussed so far:
Along with learning methods and tools for solving mathematical problems, it is important to recognize and avoid ways to make mathematical errors. This section will review some common errors.
These involve drawing a conclusion from a premise that is itself dependent on the conclusion. In other words, you are not actually proving anything. Circular reasoning often looks like deductive reasoning, but a quick examination will reveal that it’s far from it. Consider the following argument:
While this may be a simplistic example, you can see that there’s no logical procession in a circular argument.
Simply put: The fact that A implies B doesn’t not necessarily mean that B implies A. For example, “All dogs are mammals; therefore, all mammals are dogs.”
Watch out for this one. You cannot automatically assume the inverse of a given statement is true. Consider the following true statement:
If you grew up in Minnesota , you’ve seen snow.
Now, notice that the inverse of this statement is not necessarily true:
If you didn’t grow up in Minnesota , you’ve never seen snow.
This mistake (also known as inductive fallacy) can take many forms, the most common being assuming a general rule based on a specific instance: (“Bridge is a hard game; therefore, all card games are difficult.”) Be aware of more subtle forms of faulty generalizations.
It’s a mistake to assume that because two things are alike in one respect that they are necessarily alike in other ways too. Consider the faulty analogy below:
People who absolutely have to have a cup of coffee in the morning to get going are as bad as alcoholics who can’t cope without drinking.
False (or tenuous) analogies are often used in persuasive arguments.
Now that we’ve gone over some common mathematical mistakes, let’s look at some correct and effective ways to use mathematical reasoning.
Let’s look at basic logic, its operations, some fundamental laws, and the rules of logic that help us prove statements and deduce the truth. First off, there are two different styles of proofs: direct and indirect .
Whether it’s a direct or indirect proof, the engine that drives the proof is the if-then structure of a logical statement. In formal logic, you’ll see the format using the letters p and q, representing statements, as in:
If p, then q
An arrow is used to indicate that q is derived from p, like this:
This would be the general form of many types of logical statements that would be similar to: “if Joe has 5 cents, then Joe has a nickel or Joe has 5 pennies “. Basically, a proof is a flow of implications starting with the statement p and ending with the statement q. The stepping stones we use to link these statements in a logical proof on the way are called axioms or postulates , which are accepted logical tools.
A direct proof will attempt to lay out the shortest number of steps between p and q.
The goal of an indirect proof is exactly the same—it wants to show that q follows from p; however, it goes about it in a different manner. An indirect proof also goes by the names “proof by contradiction” or reductio ad absurdum . This type of proof assumes that the opposite of what you want to prove is true, and then shows that this is untenable or absurd, so, in fact, your original statement must be true.
Let’s see how this works using the isosceles triangle below. The indirect proof assumption is in bold.
Given: Triangle ABC is isosceles with B marking the vertex
Prove: Angles A and C are congruent.
Now, let’s work through this, matching our statements with our reasons.
Some math problems work on the mechanics that statements are “always”, “sometimes” and “never” true.
Example: x < x 2 for all real numbers x
We may be tempted to say that this statement is “always” true, because by choosing different values of x, like -2 and 3, we see that:
Example: For all primes x ≥ 3, x is odd.
This statement is “always” true. The only prime that is not odd is two. If we had a prime x ≥ 3 that is not odd, it would be divisible by two, which would make x not prime.
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Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.
Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.
This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations.
Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies:
Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better.
Strategies can help guide students to the solution when it is difficult ot know when to start.
The ultimate guide to problem solving techniques
Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.
Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem.
Here are 20 strategies to help students develop their problem-solving skills.
Strategies that help students understand the problem before solving it helps ensure they understand:
Following these steps leads students to the correct solution and makes the math word problem easier .
Here are five strategies to help students understand the content of the problem and identify key information.
Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.
When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed. For example, if the word problem asks how many are left, the problem likely requires subtraction. Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.
Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary. Summaries should include only the important information and be in simple terms that help contextualize the problem.
A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer. Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.
Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it. The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer. Encourage students to make a list of each step they need to take to solve the problem before getting started.
1. draw a model or diagram.
Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process. It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.
Similarly, you could draw a model to represent the objects in the problem:
This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives . When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts. The examples below show how 1st-grade students could “act out” an addition and subtraction problem:
The problem | How to act out the problem |
Gia has 6 apples. Jordan has 3 apples. How many apples do they have altogether? | Two students use counters to represent the apples. One student has 6 counters and the other student takes 3. Then, they can combine their “apples” and count the total. |
Michael has 7 pencils. He gives 2 pencils to Sarah. How many pencils does Michael have now? | One student (“Michael”) holds 7 pencils, the other (“Sarah”) holds 2 pencils. The student playing Michael gives 2 pencils to the student playing Sarah. Then the students count how many pencils Michael is left holding. |
Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution. This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.
For example,
To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71. Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.
When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved. It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.
Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve. Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.
Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense.
There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.
Here are five strategies to help students check their solutions.
For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work.
Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable. Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten. For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10. When the estimate is clear the two numbers are close. This means your answer is reasonable.
This method is particularly useful for algebraic equations. Specifically when working with variables. To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.
If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓
Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly. Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills. If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.
A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.
In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems.
Polya’s 4 steps include:
Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall.
Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom.
Here are 5 problem-solving strategies to introduce to students and use in the classroom.
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Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.
Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra.
Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.
READ MORE :
There are many different strategies for problem-solving; Here are 5 problem-solving strategies: • draw a model • act it out • work backwards • write a number sentence • use a formula
Here are 10 strategies for problem-solving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model • Act it out • Work backwards • Write a number sentence • Use a formula
1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back
Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.
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Looking for a summary on metacognition in relation to math teaching and learning?
Check out this guide featuring practical examples, tips and strategies to successfully embed metacognition across your school to accelerate math growth.
World Economic Forum published top 10 job skills for tomorrow. Our blog series has already covered the first nine, and now it is time for the tenth one; reasoning, problem-solving and ideation. As all the other ten job skills, also they are soft skills.
Of the job skills already presented in our blog series, reasoning, problem-solving and ideation is linked to analytical thinking and innovation , c omplex problem-solving , and creativity, originality and initiative .
Reasoning means the ability to proceed from hypothesis to conclusion in a logical and sensible way. The skills needed in problem-solving in turn help solve problems quickly and effectively. Problem-solving requires both an ability to correctly define a problem and finding a solution to it. Ideation in turn is often regarded as creativity, the ability to come up with new ideas and ways of doing, of testing the ideas and thus solving problems.
Problem-solving and ideation automatically brings into my mind the service design process’s double diamond; a process model for service design developed by the British Design Council (link: https://www.designcouncil.org.uk/news-opinion/what-framework-innovation-design-councils-evolved-double-diamond ).
Picture: Double diamond
The first diamond in the double diamond helps define the problem that is to be solved (discover – define), the second diamond helps to develop a solution to the problem (develop – define). In both parts of the double diamond first as much information is gathered as is possible (divergent thinking,) and after that the information is analyzed and crystallized into a solution (convergent thinking). The double diamond is widely used and in addition service design companies have further developed their own versions of it.
There is a wide range of methods and tools available for service design, and you can pick the ones that are best suited to your own work. The tools available for problem-solving and ideation support creativity and finding new solutions. If you want to learn more about the methods and tools for service design, I warmly recommend This is Service Design Doing by Marc Stickdorn et al (link: https://www.adlibris.com/fi/kirja/this-is-service-design-doing-9781491927182?gclid=Cj0KCQiAtJeNBhCVARIsANJUJ2ERp6R_g54Bx0tbQJL4pvv9qo6xrrb08B5MNm8cBnM6ZSHRisTGLogaAgMoEALw_wcB ).
In service design the problem to be solved is first defined. This helps to ensure that when ideation begins, we are indeed solving the correct problem. Although problem-solving is important, it is even more important that we are solving the correct problems. For this the discover – define phase of the double diamond are useful. Especially in the discover phase an open-minded, curious and empathetic approach is important. In the discover -define phase we are not yet finding a solution to the problem, we are concentrating on defining the problem that we will solve in the next phase.
– Anna Sahinoja
MIF Academy’s Innocamp.
March 21, 2024
Writing out intermediate steps can make it easier to solve problems.
Nick Slater for Quanta Magazine
Your grade school teacher probably didn’t show you how to add 20-digit numbers. But if you know how to add smaller numbers, all you need is paper and pencil and a bit of patience. Start with the ones place and work leftward step by step, and soon you’ll be stacking up quintillions with ease.
Problems like this are easy for humans, but only if we approach them in the right way. “How we humans solve these problems is not ‘stare at it and then write down the answer,’” said Eran Malach , a machine learning researcher at Harvard University. “We actually walk through the steps.”
That insight has inspired researchers studying the large language models that power chatbots like ChatGPT. While these systems might ace questions involving a few steps of arithmetic, they’ll often flub problems involving many steps, like calculating the sum of two large numbers. But in 2022, a team of Google researchers showed that asking language models to generate step-by-step solutions enabled the models to solve problems that had previously seemed beyond their reach. Their technique, called chain-of-thought prompting, soon became widespread, even as researchers struggled to understand what makes it work.
Now, several teams have explored the power of chain-of-thought reasoning by using techniques from an arcane branch of theoretical computer science called computational complexity theory. It’s the latest chapter in a line of research that uses complexity theory to study the intrinsic capabilities and limitations of language models. These efforts clarify where we should expect models to fail, and they might point toward new approaches to building them.
“They remove some of the magic,” said Dimitris Papailiopoulos , a machine learning researcher at the University of Wisconsin, Madison. “That’s a good thing.”
Large language models are built around mathematical structures called artificial neural networks. The many “neurons” inside these networks perform simple mathematical operations on long strings of numbers representing individual words, transmuting each word that passes through the network into another. The details of this mathematical alchemy depend on another set of numbers called the network’s parameters, which quantify the strength of the connections between neurons.
To train a language model to produce coherent outputs, researchers typically start with a neural network whose parameters all have random values, and then feed it reams of data from around the internet. Each time the model sees a new block of text, it tries to predict each word in turn: It guesses the second word based on the first, the third based on the first two, and so on. It compares each prediction to the actual text, then tweaks its parameters to reduce the difference. Each tweak only changes the model’s predictions a tiny bit, but somehow their collective effect enables a model to respond coherently to inputs it has never seen.
Researchers have been training neural networks to process language for 20 years. But the work really took off in 2017, when researchers at Google introduced a new kind of network called a transformer.
“This was proposed seven years ago, which seems like prehistory,” said Pablo Barceló , a machine learning researcher at the Pontifical Catholic University of Chile.
What made transformers so transformative is that it’s easy to scale them up — to increase the number of parameters and the amount of training data — without making training prohibitively expensive. Before transformers, neural networks had at most a few hundred million parameters; today, the largest transformer-based models have more than a trillion. Much of the improvement in language-model performance over the past five years comes from simply scaling up.
Transformers made this possible by using special mathematical structures called attention heads, which give them a sort of bird’s-eye view of the text they’re reading. When a transformer reads a new block of text, its attention heads quickly scan the whole thing and identify relevant connections between words — perhaps noting that the fourth and eighth words are likely to be most useful for predicting the 10th. Then the attention heads pass words along to an enormous web of neurons called a feedforward network, which does the heavy number crunching needed to generate the predictions that help it learn.
Real transformers have multiple layers of attention heads separated by feedforward networks, and only spit out predictions after the last layer. But at each layer, the attention heads have already identified the most relevant context for each word, so the computationally intensive feedforward step can happen simultaneously for every word in the text. That speeds up the training process, making it possible to train transformers on increasingly large sets of data. Even more important, it allows researchers to spread the enormous computational load of training a massive neural network across many processors working in tandem.
To get the most out of massive data sets, “you have to make the models really large,” said David Chiang , a machine learning researcher at the University of Notre Dame. “It’s just not going to be practical to train them unless it’s parallelized.”
However, the parallel structure that makes it so easy to train transformers doesn’t help after training — at that point, there’s no need to predict words that already exist. During ordinary operation, transformers output one word at a time, tacking each output back onto the input before generating the next word, but they’re still stuck with an architecture optimized for parallel processing.
As transformer-based models grew and certain tasks continued to give them trouble, some researchers began to wonder whether the push toward more parallelizable models had come at a cost. Was there a way to understand the behavior of transformers theoretically?
Theoretical studies of neural networks face many difficulties, especially when they try to account for training. Neural networks use a well-known procedure to tweak their parameters at each step of the training process. But it can be difficult to understand why this simple procedure converges on a good set of parameters.
Rather than consider what happens during training, some researchers study the intrinsic capabilities of transformers by imagining that it’s possible to adjust their parameters to any arbitrary values. This amounts to treating a transformer as a special type of programmable computer.
“You’ve got some computing device, and you want to know, ‘Well, what can it do? What kinds of functions can it compute?’” Chiang said.
These are the central questions in the formal study of computation. The field dates back to 1936, when Alan Turing first imagined a fanciful device , now called a Turing machine, that could perform any computation by reading and writing symbols on an infinite tape. Computational complexity theorists would later build on Turing’s work by proving that computational problems naturally fall into different complexity classes defined by the resources required to solve them.
In 2019, Barceló and two other researchers proved that an idealized version of a transformer with a fixed number of parameters could be just as powerful as a Turing machine. If you set up a transformer to repeatedly feed its output back in as an input and set the parameters to the appropriate values for the specific problem you want to solve, it will eventually spit out the correct answer.
That result was a starting point, but it relied on some unrealistic assumptions that would likely overestimate the power of transformers. In the years since, researchers have worked to develop more realistic theoretical frameworks.
One such effort began in 2021, when William Merrill , now a graduate student at New York University, was leaving a two-year fellowship at the Allen Institute for Artificial Intelligence in Seattle. While there, he’d analyzed other kinds of neural networks using techniques that seemed like a poor fit for transformers’ parallel architecture. Shortly before leaving, he struck up a conversation with the Allen Institute for AI researcher Ashish Sabharwal , who’d studied complexity theory before moving into AI research. They began to suspect that complexity theory might help them understand the limits of transformers.
“It just seemed like it’s a simple model; there must be some limitations that one can just nail down,” Sabharwal said.
The pair analyzed transformers using a branch of computational complexity theory, called circuit complexity, that is often used to study parallel computation and had recently been applied to simplified versions of transformers. Over the following year, they refined several of the unrealistic assumptions in previous work. To study how the parallel structure of transformers might limit their capabilities, the pair considered the case where transformers didn’t feed their output back into their input — instead, their first output would have to be the final answer. They proved that the transformers in this theoretical framework couldn’t solve any computational problems that lie outside a specific complexity class. And many math problems, including relatively simple ones like solving linear equations, are thought to lie outside this class.
Basically, they showed that parallelism did come at a cost — at least when transformers had to spit out an answer right away. “Transformers are quite weak if the way you use them is you give an input, and you just expect an immediate answer,” Merrill said.
Merrill and Sabharwal’s results raised a natural question — how much more powerful do transformers become when they’re allowed to recycle their outputs? Barceló and his co-authors had studied this case in their 2019 analysis of idealized transformers, but with more realistic assumptions the question remained open. And in the intervening years, researchers had discovered chain-of-thought prompting, giving the question a newfound relevance.
Merrill and Sabharwal knew that their purely mathematical approach couldn’t capture all aspects of chain-of-thought reasoning in real language models, where the wording in the prompt can be very important . But no matter how a prompt is phrased, as long as it causes a language model to output step-by-step solutions, the model can in principle reuse the results of intermediate steps on subsequent passes through the transformer. That could provide a way to evade the limits of parallel computation.
Meanwhile, a team from Peking University had been thinking along similar lines, and their preliminary results were positive. In a May 2023 paper, they identified some math problems that should be impossible for ordinary transformers in Merrill and Sabharwal’s framework, and showed that intermediate steps enabled the transformers to solve these problems.
In October, Merrill and Sabharwal followed up their earlier work with a detailed theoretical study of the computational power of chain of thought. They quantified how that extra computational power depends on the number of intermediate steps a transformer is allowed to use before it must spit out a final answer. In general, researchers expect the appropriate number of intermediate steps for solving any problem to depend on the size of the input to the problem. For example, the simplest strategy for adding two 20-digit numbers requires twice as many intermediate addition steps as the same approach to adding two 10-digit numbers.
Examples like this suggest that transformers wouldn’t gain much from using just a few intermediate steps. Indeed, Merrill and Sabharwal proved that chain of thought only really begins to help when the number of intermediate steps grows in proportion to the size of the input, and many problems require the number of intermediate steps to grow much larger still.
The thoroughness of the result impressed researchers. “They really pinned this down,” said Daniel Hsu , a machine learning researcher at Columbia University.
Merrill and Sabharwal’s recent work indicates that chain of thought isn’t a panacea — in principle, it can help transformers solve harder problems, but only at the cost of a lot of computational effort.
“We’re interested in different ways of getting around the limitations of transformers with one step,” Merrill said. “Chain of thought is one way, but this paper shows that it might not be the most economical way.”
Still, researchers caution that this sort of theoretical analysis can only reveal so much about real language models. Positive results — proofs that transformers can in principle solve certain problems — don’t imply that a language model will actually learn those solutions during training.
And even results that address the limitations of transformers come with caveats: They indicate that no transformer can solve certain problems perfectly in all cases. Of course, that’s a pretty high bar. “There might be special cases of the problem that it could handle just fine,” Hsu said.
Despite these caveats, the new work offers a template for analyzing different kinds of neural network architectures that might eventually replace transformers. If a complexity theory analysis suggests that certain types of networks are more powerful than others, that would be evidence that those networks might fare better in the real world as well.
Chiang also stressed that research on the limitations of transformers is all the more valuable as language models are increasingly used in a wide range of real-world applications, making it easy to overestimate their abilities.
“There’s actually a lot of things that they don’t do that well, and we need to be very, very cognizant of what the limitations are,” Chiang said. “That’s why this kind of work is really important.”
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A review of the development and future challenges of case-based reasoning.
2. basic framework and concept of cbr, 3. development of cbr key technologies, 3.1. case representation, 3.2. similarity measure and case retrieval, 3.2.1. similarity measure, 3.2.2. case retrieval.
4. application fields of cbr, 4.1. diagnosis, 4.2. prediction, 4.3. design and planning, 4.4. decision support, 4.5. recommendation system, 4.6. other applications, 5. summary and challenge, author contributions, conflicts of interest.
Click here to enlarge figure
Method | Element | Representation | Limitation | ||
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Frame | slot: facet: value | <Frame name> | Low reasoning efficiency; Hard to track and control. | ||
slot 1: | facet 1 | value 1 , value 1 , | |||
facet 1 | value 1 , value 1 , | ||||
slot n: | facet n | value n , value n , | |||
facet n | value n , value n , | ||||
constraint: | constraint condition | ||||
Object-Oriented | CLASS::=<ID, DS, MS, MI> ID: Identifier DS: Data Structure MS: Method Set MI: Message Interface | class <name>[:<Superclass>] | |||
[<Class variable name>] | |||||
Structure | |||||
<Static structure description of an object> | |||||
Method | |||||
<Definition of an operation on an object> | |||||
Restraint | |||||
<Restricted condition> | |||||
END | |||||
Production Rule | <production>::= <precondition> <conclusion> | P→Q IF P THEN Q (CF = [0, 1]) CF: Certainty Factor | Low efficiency; Unable to express structured knowledge | ||
Semantic Nets | (Node1, Arc, Node2) Semantic Relation | AKO: A-Kind-Of | Non-rigidity; Low reasoning efficiency; Knowledge access complexity | ||
Predicate-Based | Predicate (Constant/Variate/Function) Conjunctions Quantifier | Predicate Formula | Cannot represent uncertain knowledge; Combinatorial explosion; Low efficiency |
Relation | Function | ||||
---|---|---|---|---|---|
⇔ | x and y are similar | ⇔ | x and y are exactly similar | ||
⇔ | x and y are dissimilar | ⇔ | x and y are exactly dissimilar | ||
⇔ | x is at least as similar to y as x to z | ⇔ | x and y are partly similar |
Deep Learning | CBR |
---|---|
Data, experience, and knowledge are all examples | Cases |
It is about learning knowledge | It is about learning knowledge |
General rules and laws are generated | Specific solutions are generated |
Technology | Methodology |
Unsupervised learning possible | Unsupervised problem solving cannot be done |
Eager learners | Lazy learners |
Results are not precise or certain | Results are not precise or certain |
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Yan, A.; Cheng, Z. A Review of the Development and Future Challenges of Case-Based Reasoning. Appl. Sci. 2024 , 14 , 7130. https://doi.org/10.3390/app14167130
Yan A, Cheng Z. A Review of the Development and Future Challenges of Case-Based Reasoning. Applied Sciences . 2024; 14(16):7130. https://doi.org/10.3390/app14167130
Yan, Aijun, and Zijun Cheng. 2024. "A Review of the Development and Future Challenges of Case-Based Reasoning" Applied Sciences 14, no. 16: 7130. https://doi.org/10.3390/app14167130
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A research team from the Beijing Academy of Artificial Intelligence and China University of Mining & Technology has proposed a scalable dataset for programmatic mathematical reasoning called InfinityMath. According to the authors, InfinityMath is supposed to decouple numeric values from problems stated in mathematics. This way, creating a huge, diverse dataset will require a manageable amount of computational resources. The dataset was created from seven high-quality math sources. It has over 101,380 data points. This makes it quite a comprehensive tool for enhancing the reasoning ability of artificial intelligence models.
The methodology of InfinityMath is multistep for maximum scalability and logical consistency. Masking numerical values of math problems creates generic templates that provide a base for generating problem-solving programs. These are then taken as general templates for developing programs that do not refer to specific numbers, logically following the same reasoning procedure for all possible numerical variations. It can efficiently scale data and improve the resiliency of AI models across different mathematical challenges. Such programs could be generated with sophisticated language models like GPT-4 to reduce potential errors and improve overall quality.
The models fine-tuned with the InfinityMath dataset performed quite well across several benchmarks. For example, aided by the InfinityMath dataset, the Llama2 model showed sensational accuracy improvements in the GSM8K dataset at 316.44% and in the MATH dataset at 1067.6%. Another model fine-tuned on this dataset was CodeLlama, which also showed huge improvements: 120.58% in SVAMP and 1118.09% in SimulEq. These results show that, at the very least, InfinityMath can increase AI models’ accuracy and robustness and improve their reliability in solving various mathematical problems. This consistency was also ahead regarding logical outcomes due to numerical variations; traditional datasets often lack performance.
Therefore, The InfinityMath effect extends beyond mere numerical accuracy to strike at perhaps the most fundamental feature of mathematical reasoning. The authors performed strict, improved evaluations with existing test sets, such as GSM8K+ and MATH+, differing only in the numerical values. Models trained on InfinityMath showed higher performance in logical consistency than any other dataset in accuracy and model efficacy. This success underlines the role played by InfinityMath in further pushing the frontiers of mathematical reasoning and scaling and making an effective solution available to a very large class of AI models.
In other words, InfinityMath is a major improvement in mathematical reasoning, solving two major challenges: scalability and logical consistency. The dataset was curated by a dedicated research team from the Beijing Academy of Artificial Intelligence and the China University of Mining & Technology to ensure that a robust and highly extensible solution could ultimately allow AI models to solve extremely complex mathematical problems. In this case, the InfinityMath process not only separates numerical values from solving processes but also makes constructing a large, highly diversified dataset more efficient to enhance the accuracy and reliability of the AI models. These results thus enable gains in improvement to be witnessed with multiple benchmark-related performances. Therefore, this dataset could further improve AI and its applications in various fields.
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Module 7: Thinking, Reasoning, and Problem-Solving. This module is about how a solid working knowledge of psychological principles can help you to think more effectively, so you can succeed in school and life. You might be inclined to believe that—because you have been thinking for as long as you can remember, because you are able to figure ...
Fluency, reasoning and problem solving are central strands of mathematical competency, as recognized by the National Council of Teachers of Mathematics (NCTM) and the National Research Council's report 'Adding It Up'. They are key components to the Standards of Mathematical Practice, standards that are interwoven into every mathematics ...
Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue. The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything ...
REASONING 5. The Problem of Induction 95 Steven A. Sloman David A. Lagnado 6. Analogy 117 Keith J. Holyoak 7. Causal Learning 143 Marc J. Buehner Patricia W. Cheng 8. ... PROBLEM SOLVING AND COMPLEX LEARNING 14. Problem Solving 321 Laura R. Novick Miriam Bassok 15. Creativity 351 Robert J. Sternberg Todd I. Lubart James C. Kaufman
A logical puzzle is a problem that can be solved through deductive reasoning. This page gives a summary of the types of logical puzzles one might come across and the problem-solving techniques used to solve them. One of the simplest types of logical puzzles is a syllogism. In this type of puzzle, you are given a set of statements, and you are required to determine some truth from those statements.
Problem-solving skills are essential in our daily lives. The video explains different problem-solving methods, including trial and error, algorithm strategy, and heuristics. It also discusses concepts like means-end analysis, working backwards, fixation, and insight. These techniques help us tackle both well-defined and ill-defined problems ...
Additional Problem Solving Strategies:. Abstraction - refers to solving the problem within a model of the situation before applying it to reality.; Analogy - is using a solution that solves a similar problem.; Brainstorming - refers to collecting an analyzing a large amount of solutions, especially within a group of people, to combine the solutions and developing them until an optimal ...
Problem-solving is a vital skill for coping with various challenges in life. This webpage explains the different strategies and obstacles that can affect how you solve problems, and offers tips on how to improve your problem-solving skills. Learn how to identify, analyze, and overcome problems with Verywell Mind.
Abstract. This chapter provides a revised review of the psychological literature on reasoning and problem solving. Four classes of deductive reasoning are presented, including rule (mental logic) theories, semantic (mental model) theories, evolutionary theories, and heuristic theories. Major developments in the study of reasoning are also ...
The first step in solving a problem is understanding what that problem actually is. You need to be sure that you're dealing with the real problem - not its symptoms. For example, if performance in your department is substandard, you might think that the problem lies with the individuals submitting work. However, if you look a bit deeper, the ...
the psychology of thinking is concerned with complex mental behaviours, such as problem- solving, reasoning, decision-making, and becoming an expert. A good understanding of basic cognition is very useful in understanding the psychology of thinking, but it is not neces-sary. In other words, if you are reading this book as part of a course on ...
TRIZ: A Russian-developed problem solving technique that values logic, analysis, and forecasting over intuition or soft reasoning. TRIZ (translated to "theory of inventive problem solving" or TIPS in English) is a systematic approach to defining and identifying an inventive solution to difficult problems.
Finding a suitable solution for issues can be accomplished by following the basic four-step problem-solving process and methodology outlined below. Step. Characteristics. 1. Define the problem. Differentiate fact from opinion. Specify underlying causes. Consult each faction involved for information. State the problem specifically.
Logical Problem-Solving. The definition of logical problem solving is relatively self-explanatory. Logical reasoning, or logic, is used in order to work through a problem and find a solution. If ...
Problem-solving: Problem-solving is perhaps the most important skill that critical thinkers can possess. The ability to solve issues and bounce back from conflict is what helps you succeed, be a leader, and effect change. ... Over time, developing your logic and reasoning will strengthen your critical thinking skills. Next steps. Learn tips and ...
In this video you will learn to define the terms and concepts problem solving and employ inductive and deductive reasoning in problem solving. References: Au...
7. Solution evaluation. 1. Problem identification. The first stage of any problem solving process is to identify the problem (s) you need to solve. This often looks like using group discussions and activities to help a group surface and effectively articulate the challenges they're facing and wish to resolve.
The problem-solving process typically includes the following steps: Identify the issue: Recognize the problem that needs to be solved. Analyze the situation: Examine the issue in depth, gather all relevant information, and consider any limitations or constraints that may be present. Generate potential solutions: Brainstorm a list of possible ...
Problem Solving Reasoning is a logical reasoning part where candidates will be given various questions and they need to perform various operations such as addition, division, greater than, lesser than, etc are interchanged or substituted to find the correct answer. Almost all the government examinations ask questions on the problem solving reasoning section.
Explaining mathematical reasoning and problem solving by using a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and concrete models can help students understand the problem better by making it more concrete and approachable. Let's try another one.
Here are five strategies to help students check their solutions. 1. Use the Inverse Operation. For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7.
In insight problem-solving, the cognitive processes that help you solve a problem happen outside your conscious awareness. 4. Working backward. Working backward is a problem-solving approach often ...
Reasoning means the ability to proceed from hypothesis to conclusion in a logical and sensible way. The skills needed in problem-solving in turn help solve problems quickly and effectively. Problem-solving requires both an ability to correctly define a problem and finding a solution to it. Ideation in turn is often regarded as creativity, the ...
In general, researchers expect the appropriate number of intermediate steps for solving any problem to depend on the size of the input to the problem. For example, the simplest strategy for adding two 20-digit numbers requires twice as many intermediate addition steps as the same approach to adding two 10-digit numbers.
Given a problem and a set of premises, problem-solving reduces to searching for a proof tree whose root node is labelled by a solution of the problem and whose leaf nodes are labelled by premises or axioms. In the case of Horn clauses, problem-solving search can be performed by reasoning forwards from the premises or backwards from the problem ...
Finally, reflection is a critical component of problem-solving in QA. After a solution has been implemented, take time to analyze the process that led to it. Use logical reasoning to evaluate what ...
Case-based reasoning (CBR), which is based on the cognitive assumption that similar problems have similar solutions, is an important problem-solving and learning method in the field of artificial intelligence (AI). In this article, the development of CBR is reviewed, and the major challenges of CBR are summarized. The paper is organized into four parts.
One primary driver for artificial intelligence research in mathematical reasoning is that it may further increase model understanding and problem-solving abilities on complex mathematical problems. Applications such as these can be very important in education, finance, and technology—fields dependent on the accuracy of solutions and the speed at which problems are solved.