problem solving in modeling

Common Problem-Solving Models & How to Use Them

Problem – solving models are step-by-step processes that provide a framework for addressing challenges. Problems arise in every facet of life. From work. to home. to friends and family, problems and conflicts can make life difficult and interfere with our physical and mental well-being. Understanding how to approach problems when they arise and implementing problem-solving techniques can make the journey through a problem less onerous on ourselves and those around us.

By building a structured problem-solving process, you can begin to build muscle memory by repeatedly practicing the same approach, and eventually, you may even begin to find yourself solving complex problems . Building a problem-solving model for each of the situations where you may encounter a problem can give you a path forward, even when the most difficult of problems arise.

This article will explore the concept of problem-solving models and dive into examples of such models and how to use them. It will also outline the benefits of implementing a problem-solving model in each area of life and why these problem-solving methods can have a large impact on your overall well-being. The goal of this article is to help you identify effective problem-solving strategies and develop critical thinking to generate solutions for any problem that comes your way.

Problem-Solving Model Defined

The first step in creating a problem-solving plan is to understand what we mean when we say problem-solving models. A problem-solving model is a step-by-step process that helps a team identify and effectively solve problems that they may encounter. This problem-solving approach gives the team the muscle memory and guide to address a conflict and resolve disputes quickly and effectively.

There are common problem-solving models that many teams have implemented, but there is also the freedom to shape a method to fit the needs of a specific situation. These models often rely on various problem-solving techniques to identify the root cause of the issue and find the best solution. This article will explore some common problem-solving models as well as general problem-solving techniques to help a team engage with and solve problems effectively.

Benefits of Implementing Problem-Solving Models

Before we discuss the exact models for problem-solving, it can be helpful to discuss why problem-solving models are beneficial in the first place. There are a variety of benefits to having a plan in place when a problem arises, but a few important benefits are listed below.

Guide Posts

When a team encounters a problem and has a guide for how to approach and solve the problem, it can be a relief to know that they have a process to fall back on when the issue cannot be resolved quickly from the beginning. A problem-solving strategy will serve as a guide for the parties to know which steps to take next and how to identify the appropriate solution.

It can also clarify when the issue needs to stay within the team, and when the issue needs to be escalated to someone in a position with more authority. It can also help the entire team solve complex problems without creating an issue out of the way the team solves the problem. It gives the team a blueprint to work from and encourages them to find a good solution.

Creative Solutions That Last

When the team or family has a way to fall back on to solve a problem, it takes some of the pressure off of coming up with the process and allows the parties to focus on identifying the relevant information and coming up with various potential solutions to the issue. By using a problem-solving method, the parties can come up with different solutions and find common ground with the best solution. This can be stifled if the team is too focused on figuring out how to solve the problem.

Additionally, the solutions that the parties come up with through problem-solving tools will often address the root cause of the issue and stop the team from having to revisit the same problem over and over again. This can lead to overall productivity and well-being and help the team continue to output quality work. By encouraging collaboration and creativity, a problem-solving technique will often keep solving problems between the parties moving forward and possibly even address them before they show up.

Common Models to Use in the Problem-Solving Process

Several models can be applied to a complex problem and create possible solutions. These range from common and straightforward to creative and in-depth to identify the most effective ways to solve a problem. This section will discuss and break down the problem-solving models that are most frequently used.

Standard Problem-Solving Process

When you search for a problem-solving technique, chances are you will find the standard model for saving problems. This model identifies and uses several important steps that will often be used in other models as well, so it can be helpful to begin the model-building process with an understanding of this model as a base. Other models often draw from this process and adapt one or more of the steps to help create additional options. Each of these steps works to accomplish a specific goal in furtherance of a solution.

Define the Problem

The first step in addressing a problem is to create a clear definition of the issue at hand. This will often require the team to communicate openly and honestly to place parameters around the issue. As the team defines the problem, it will be clear what needs to be solved and what pieces of the conflict are ancillary to the major issue. It helps to find the root causes of the issue and begin a process to address that rather than the symptoms of the problem. The team can also create a problem statement, which outlines the parameters of the problem and what needs to be fixed.

In addition to open and honest communication, other techniques can help to identify the root cause and define the problem. This includes a thorough review of the processes and steps that are currently used in the task and whether any of those steps are directly or indirectly causing the problem.

This includes reviewing how tasks are done, how communication is shared, and the current partners and team members that work together to identify if any of those are part of the issue. It is also the time to identify if some of the easy fixes or new tools would solve the problem and what the impact would be.

It is also important to gain a wide understanding of the problem from all of the people involved. Many people will have opinions on what is going on, but it is also important to understand the facts over the opinions that are affecting the problem. This can also help you identify if the problem is arising from a boundary or standard that is not being met or honored. By gathering data and understanding the source of the problem, the process of solving it can begin.

Generate Solutions

The next step in the basic process is to generate possible solutions to the problem. At this step, it is less important to evaluate how each of the options will play out and how they may change the process and more important to identify solutions that could address the issue. This includes solutions that support the goals of the team and the task, and the team can also identify short and long-term solutions.

The team should work to brainstorm as many viable solutions as possible to give them the best options to consider moving forward. They cannot pick the first solution that is proposed and consider it a successful problem-solving process.

Evaluate and Select

After a few good options have been identified, the next step is to evaluate the options and pick the most viable option that also supports the goals of the team or organization. This includes looking at each of the possible solutions and determining how they would either encourage or hinder the goals and standards of the team. These should evaluated without bias toward the solution proposed or the person putting forward the solution. Additionally, the team should consider both actual outcomes that have happened in the past and predicted instances that may occur if the solution is chosen.

Each solution should be evaluated by considering if the solution would solve the current problem without causing additional issues, the willingness of the team to buy in and implement the solution, and the actual ability of the team to implement the solution.

Participation and honesty from all team members will make the process go more smoothly and ensure that the best option for everyone involved is selected. Once the team picks the option they would like to use for the specific problem, they should clearly define what the solution is and how it should be implemented. There should also be a strategy for how to evaluate the effectiveness of the solution.

Implement the Solution and Follow Up

Once a solution is chosen, a team will often assume that the work of solving problems is complete. However, the final step in the basic model is an important step to determine if the matter is resolved or if additional options are needed. After the solution has been implemented by the team, the members of the team must provide feedback and identify any potential obstacles that may have been missed in the decision-making process.

This encourages long-term solutions for the problem and helps the team to continue to move forward with their work. It also gives the team a sense of ownership and an example of how to evaluate an idea in the future.

If the solution is not working the way that it should, the team will often need to adapt the option, or they may get to the point where they scrap the option and attempt another. Solving a problem is not always a linear process, and encouraging reform and change within the process will help the team find the answer to the issues that they face.

GROW Method

Another method that is similar to the standard method is the G.R.O.W. method. This method has very similar steps to the standard method, but the catchiness of the acronym helps a team approach the problem from the same angle each time and work through the method quickly.

The first step in the method is to identify a goal, which is what the “g” stands for in “grow.” To establish a goal, the team will need to look at the issues that they are facing and identify what they would like to accomplish and solve through the problem-solving process. The team will likely participate in conversations that identify the issues that they are facing and what they need to resolve.

The next step is to establish the current reality that the group is facing. This helps them to determine where they currently are and what needs to be done to move them forward. This can help the group establish a baseline for where they started and what they would like to change.

The next step is to find any obstacles that may be blocking the group from achieving their goal. This is where the main crux of the issues that the group is facing will come out. This is also helpful in giving the group a chance to find ways around these obstacles and toward a solution.

Way Forward

After identifying the obstacles and potential ways to avoid them, the group will then need to pick the best way to move forward and approach their goal together. Here, they will need to create steps to move forward with that goal.

Divide and Conquer

Another common problem-solving method is the divide-and-conquer method. Here, instead of the entire team working through each step of the process as a large group, they split up the issue into smaller problems that can be solved and have individual members or small groups work through the smaller problems. Once each group is satisfied with the solution to the problem, they present it to the larger group to consider along with the other options.

This process can be helpful if there is a large team attempting to solve a large and complex problem. It is also beneficial because it can be used in teams with smaller, specialized teams within it because it allows each smaller group to focus on what they know best.

However, it does encourage the parties to shy away from collaboration on the overall issue, and the different solutions that each proposes may not be possible when combined and implemented.

For this reason, it is best to use this solution when approaching complex problems with large teams and the ability to combine several problem-solving methods into one.

Six Thinking Hats

The Six Thinking Hats theory is a concept designed for a team with a lot of differing conflict styles and problem-solving techniques. This method was developed to help sort through the various techniques that people may use and help a team find a solution that works for everyone involved. It helps to organize thinking and lead the conversation to the best possible solution.

Within this system, there are six different “hats” that identify with the various aspects of the decision-making process: the overall process, idea generation, intuition and emotions, values, information gathering, and caution or critical thinking. The group agrees to participate in the process by agreeing on which of the hats the group is wearing at a given moment. This helps set parameters and expectations around what the group is attempting to achieve at any moment.

This system is particularly good in a group with different conflict styles or where people have a hard time collecting and organizing their thoughts. It can be incredibly beneficial for complex problems with many moving parts. It can also help groups identify how each of the smaller sections relates to the big picture and help create new ideas to answer the overall problem.

However, it can derail if the group focuses too heavily or for too long on one of the “hats.” The group should ensure that they have a facilitator to guide them through the process and ensure that each idea and section is considered adequately.

Trial and Error

The trial and error process takes over the evaluation and selection process and instead chooses to try out each of the alternatives to determine what the best option would be. It allows the team to gather data on each of the options and how they apply practically. It also provides the ability for the team to have an example of each possible answer to help a decision-maker determine what the best option is.

Problem-solving methods that focus on trial and error can be helpful when a team has a simple problem or a lot of time to test potential solutions, gather data, and determine an answer to the issue.

It can also be helpful when the team has a sense of the best guess for a solution but wants to test it out to determine if the data supports that option, or if they have several viable options and would like to identify the best one. However, it can be incredibly time-consuming to test each of the options and evaluate how they went. Time can often be saved by evaluating each option and selecting the best to test.

Ask Good Questions

One of the best ways to work through any of the problem-solving models is to ask good questions. This will help the group find the issue at the heart of the problem and address that issue rather than the symptoms. The best questions will also help the group find viable solutions and pick the solution that the group can use to move forward. The more creative the questions , the more likely that they will produce innovative solutions.

Take a Step Back

Occasionally, paying attention to a problem too much can give the group tunnel vision and harm the overall processes that the group is using. Other times, the focus can lead to escalations in conflict. When this happens, it can be helpful to set aside the problem and give the group time to calm down. Once they have a chance to reconsider the options and how they apply, they can approach the issue with a new sense of purpose and determination. This can lead to additional creative solutions that may help the group find a new way forward.

Final Thoughts

Problem-solving can be a daunting part of life. However, with a good problem-solving method and the right techniques, problems can be addressed well and quickly. Applying some of these options outlined in this article can give you a head start in solving your next problem and any others that arise.

To learn more about problem-solving models, problem-solving activities, and more, contact ADR Times !

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35 problem-solving techniques and methods for solving complex problems

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How do you identify problems?

How do you identify the right solution.

Tips for more effective problem-solving

Complete problem-solving methods

Problem-solving techniques to identify and analyze problems
Problem-solving techniques for developing solutions

Problem-solving warm-up activities

Closing activities for a problem-solving process.

Before you can move towards finding the right solution for a given problem, you first need to identify and define the problem you wish to solve.

Here, you want to clearly articulate what the problem is and allow your group to do the same. Remember that everyone in a group is likely to have differing perspectives and alignment is necessary in order to help the group move forward.

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 scary for people to stand up and contribute, especially if the problems or challenges are emotive or personal in nature. Be sure to try and create a psychologically safe space for these kinds of discussions.

Remember that problem analysis and further discussion are also important. Not taking the time to fully analyze and discuss a challenge can result in the development of solutions that are not fit for purpose or do not address the underlying issue.

Successfully identifying and then analyzing a problem means facilitating a group through activities designed to help them clearly and honestly articulate their thoughts and produce usable insight.

With this data, you might then produce a problem statement that clearly describes the problem you wish to be addressed and also state the goal of any process you undertake to tackle this issue.

Finding solutions is the end goal of any process. Complex organizational challenges can only be solved with an appropriate solution but discovering them requires using the right problem-solving tool.

After you’ve explored a problem and discussed ideas, you need to help a team discuss and choose the right solution. Consensus tools and methods such as those below help a group explore possible solutions before then voting for the best. They’re a great way to tap into the collective intelligence of the group for great results!

Remember that the process is often iterative. Great problem solvers often roadtest a viable solution in a measured way to see what works too. While you might not get the right solution on your first try, the methods below help teams land on the most likely to succeed solution while also holding space for improvement.

Every effective problem solving process begins with an agenda . A well-structured workshop is one of the best methods for successfully guiding a group from exploring a problem to implementing a solution.

In 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!

Tips for more effective problem solving

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!

Clearly define the problem

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.

Don’t jump to conclusions

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.

Try different approaches

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.

Don’t take it personally

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.

Get the right people in the room

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.

Document everything

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!

Bring a facilitator

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!

Develop your problem-solving skills

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!

Design a great agenda

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!

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.

Six Thinking Hats
Lightning Decision Jam
Problem Definition Process
Discovery & Action Dialogue

Design Sprint 2.0

Open Space Technology

1. Six Thinking Hats

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.

2. Lightning Decision Jam

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

3. Problem Definition Process

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.

4. The 5 Whys

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.

5. World Cafe

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.

6. Discovery & Action Dialogue (DAD)

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.

7. 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.

8. Open space technology

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

Techniques to identify and analyze problems

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!

The Creativity Dice
Fishbone Analysis
Problem Tree
SWOT Analysis
Agreement-Certainty Matrix
The Journalistic Six
LEGO Challenge
What, So What, Now What?
Journalists

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.

10. The Creativity Dice

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.

11. Fishbone Analysis

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.

12. Problem Tree

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.

13. SWOT Analysis

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.

14. Agreement-Certainty Matrix

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.

16. Speed Boat

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.

17. The Journalistic Six

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.

18. LEGO Challenge

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.

19. What, So What, Now What?

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!

20. Journalists

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.

Problem-solving techniques for developing solutions

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 narrow down to the correct solution.

Use these problem-solving techniques when you want to help your team find consensus, compare possible solutions, and move towards taking action on a particular problem.

Improved Solutions
Four-Step Sketch
15% Solutions
How-Now-Wow matrix
Impact Effort Matrix

21. Mindspin

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.

22. Improved Solutions

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.

23. Four Step Sketch

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

24. 15% Solutions

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.

25. How-Now-Wow Matrix

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.

26. Impact and Effort Matrix

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.

27. Dotmocracy

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.

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.

Check-in/Check-out
Doodling Together
Show and Tell
Constellations
Draw a Tree

28. Check-in / Check-out

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.

29. Doodling Together

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.

30. Show and Tell

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.

31. Constellations

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.

32. Draw a Tree

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.

One Breath Feedback
Who What When Matrix
Response Cards

How do I conclude a problem-solving process?

All good things must come to an end. With the bulk of the work done, it can be tempting to conclude your workshop swiftly and without a moment to debrief and align. This can be problematic in that it doesn’t allow your team to fully process the results or reflect on the process.

At the end of an effective session, your team will have gone through a process that, while productive, can be exhausting. It’s important to give your group a moment to take a breath, ensure that they are clear on future actions, and provide short feedback before leaving the space.

The primary purpose of any problem-solving method is to generate solutions and then implement them. Be sure to take the opportunity to ensure everyone is aligned and ready to effectively implement the solutions you produced in the workshop.

Remember that every process can be improved and by giving a short moment to collect feedback in the session, you can further refine your problem-solving methods and see further success in the future too.

33. One Breath Feedback

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.

34. Who What When Matrix

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.

35. Response cards

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.

Save time and effort discovering the right solutions

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!

Over to you

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 creative 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!

thank you very much for these excellent techniques

Certainly wonderful article, very detailed. Shared!

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Article • 8 min read

The FOCUS Model

A simple, efficient problem-solving approach.

By the Mind Tools Content Team

Are your business processes perfect, or could you improve them?

In an ever-changing world, nothing stays perfect for long. To stay ahead of your competitors, you need to be able to refine your processes on an ongoing basis, so that your services remain efficient and your customers stay happy.

This article looks the FOCUS Model – a simple quality-improvement tool that helps you do this.

About the Model

The FOCUS Model, which was created by the Hospital Corporation of America (HCA), is a structured approach to Total Quality Management (TQM) , and it is widely used in the health care industry.

The model is helpful because it uses a team-based approach to problem solving and to business-process improvement, and this makes it particularly useful for solving cross-departmental process issues. Also, it encourages people to rely on objective data rather than on personal opinions, and this improves the quality of the outcome.

It has five steps:

F ind the problem.
O rganize a team.
C larify the problem.
U nderstand the problem.
S elect a solution.

Applying the FOCUS Model

Follow the steps below to apply the FOCUS Model in your organization.

Step 1: Find the Problem

The first step is to identify a process that needs to be improved. Process improvements often follow the Pareto Principle , where 80 percent of issues come from 20 percent of problems. This is why identifying and solving one real problem can significantly improve your business, if you find the right problem to solve.

According to a popular analogy, identifying problems is like harvesting apples. At first, this is easy – you can pick apples up from the ground and from the lower branches of the tree. But the more fruit you collect, the harder it becomes. Eventually, the remaining fruit is all out of reach, and you need to use a ladder to reach the topmost branches.

Start with a simple problem to get the team up to speed with the FOCUS method. Then, when confidence is high, turn your attention to more complex processes.

If the problem isn't obvious, use these questions to identify possible issues:

What would our customers want us to improve?
How can we improve quality ?
What processes don't work as efficiently as they could?
Where do we experience bottlenecks in our processes?
What do our competitors or comparators do that we could do?
What frustrates and irritates our team?
What might happen in the future that could become a problem for us?

If you have several problems that need attention, list them all and use Pareto Analysis , Decision Matrix Analysis , or Paired Comparison Analysis to decide which problem to address first. (If you try to address too much in one go, you'll overload team members and cause unnecessary stress.)

Step 2: Organize a Team

Your next step is to assemble a team to address the problem.

Where possible, bring together team members from a range of disciplines – this will give you a broad range of skills, perspectives, and experience to draw on.

Select team members who are familiar with the issue or process in hand, and who have a stake in its resolution. Enthusiasm for the project will be greatest if people volunteer for it, so emphasize how individuals will benefit from being involved.

If your first choice of team member isn't available, try to appoint someone close to them, or have another team member use tools like Perceptual Positioning and Rolestorming to see the issue from their point of view.

Keep in mind that a diverse team is more likely to find a creative solution than a group of people with the same outlook.

Step 3: Clarify the Problem

Before the team can begin to solve the problem, you need to define it clearly and concisely.

According to " Total Quality Management for Hospital Nutrition Services ," a key text on the FOCUS Model, an enthusiastic team may be keen to attack an "elephant-sized" problem, but the key to success is to break it down into "sushi-sized" pieces that can be analyzed and solved more easily.

Use the Drill Down technique to break big problems down into their component parts. You can also use the 5 Whys Technique , Cause and Effect Analysis , and Root Cause Analysis to get to the bottom of a problem.

Record the details in a problem statement, which will then serve as the focal point for the rest of the exercise ( CATWOE can help you do this effectively.) Focus on factual events and measurable conditions such as:

Who does the problem affect?
What has happened?
Where is it occurring?
When does it happen?

The problem statement must be objective, so avoid relying on personal opinions, gut feelings, and emotions. Also, be on guard against "factoids" – statements that appear to be facts, but that are really opinions that have come to be accepted as fact.

Step 4: Understand the Problem

Once the problem statement has been completed, members of the team gather data about the problem to understand it more fully.

Dedicate plenty of time to this stage, as this is where you will identify the fundamental steps in the process that, when changed, will bring about the biggest improvement.

Consider what you know about the problem. Has anyone else tried to fix a similar problem before? If so, what happened, and what can you learn from this?

Use a Flow Chart or Swim Lane Diagram to organize and visualize each step; this can help you discover the stage at which the problem is happening. And try to identify any bottlenecks or failures in the process that could be causing problems.

As you develop your understanding, potential solutions to the problem may become apparent. Beware of jumping to "obvious" conclusions – these could overlook important parts of the problem, and could create a whole new process that fails to solve the problem.

Generate as many possible solutions as you can through normal structured thinking, brainstorming , reverse brainstorming , and Provocation . Don't criticize ideas initially – just come up with lots of possible ideas to explore.

Step 5: Select a Solution

The final stage in the process is to select a solution.

Use appropriate decision-making techniques to select the most viable option. Decision Trees , Paired Comparison Analysis , and Decision Matrix Analysis are all useful tools for evaluating your options.

Once you've selected an idea, use tools such as Risk Analysis , "What If" Analysis , and the Futures Wheel to think about the possible consequences of moving ahead, and make a well-considered go/no-go decision to decide whether or not you should run the project.

People commonly use the FOCUS Model in conjunction with the Plan-Do-Check-Act cycle. Use this approach to implement your solutions in a controlled way.

The FOCUS Model is a simple quality-improvement tool commonly used in the health care industry. You can use it to improve any process, but it is particularly useful for processes that span different departments.

The five steps in FOCUS are as follows:

People often use the FOCUS Model in conjunction with the Plan-Do-Check-Act cycle, which allows teams to implement their solution in a controlled way.

Bataldan, P. (1992). 'Building Knowledge for Improvement: an Introductory Guide to the Use of FOCUS-PDCA,' Nashville: TN Quality Resource Group, Hospital Corporation of America.

Schiller, M., Miller-Kovach, M., and Miller-Kovach, K. (1994). 'Total Quality Management for Hospital Nutrition Services,' Aspen Publishers Inc. Available here .

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10 Modeling Problem Solving

We’ve discussed in previous chapters how part of a tutor’s task is to model good learning habits. When tutors are organized, use good time management, and leverage resources, we demonstrate the skills that students can use to be successful learners.

Problem-solving is an additional skill that tutors model for students. An organized and- intentional problem-solving approach helps us to efficiently work through challenges, and many of us effectively problem solve without much thought given to our approach. 1 However, it makes sense to take a step back and do our best to model problem-solving best-practices. Remember that repeated demonstration of a tutor’s problem-solving strategies can help students learn from our example.

We know the tutor’s role is not to solve a student’s problem for them. How do we model good problem-solving, without actually solving the problem ourselves? It’s tricky, but not impossible. We can empower students to work their way through any problem by asking good questions and walking them through the steps of the process.

The Rational Problem-Solving Process

Problem-solving is something many of us have taught ourselves through practice. However, there are many scholars and professionals who have examined and broken down effective problem-solving strategies into a series of logical steps. 2 We can check our own process by reflecting on what has been written about best-practices in problem-solving, and maybe make changes to be more consistent and effective. This can better prepare tutors to guide a student through the process when we apply it in a tutoring session.

Step 1: Define the Problem

It may seem obvious to state that the first step in solving a problem is to notice that we have a problem. Unless we take time to understand precisely what is wrong, however, we may find ourselves creating a solution that doesn’t actually fix anything. It’s very common to dive straight into devising a solution only to find that we’ve solved the wrong problem. Alternatively, we might develop a solution only to discover that the real problem is bigger than we thought.

A good practice for starting out is to try to define the problem in words. By writing or stating a problem definition, we’re challenged to identify the root cause, and this information can guide us in developing effective solutions.

In a tutoring session, sometimes the problem can take a variety of forms. The problem could be:

the literal problem given in a student’s homework assignment (a word problem in math, or a case study in biology, for example.)
a lack of clarity in assignment instructions.
the student not having a strategy for planning a project or starting a paper.
the student lacking confidence to tackle their homework or study independently

Keep in mind that the form the “problem” takes will change based on the student’s needs and goals. If the problem is that the student doesn’t understand something, the first step is to identify precisely what they don’t understand. If the problem is that something is missing, then understanding exactly what necessary parts are missing is the first step.

In a tutoring session this may mean asking the student to start the process, or begin describing the concept from the beginning, until they reach the point where things become unclear. Together, you can determine where the gaps are, and begin to develop a problem definition.

Step 2: Pull from Existing Knowledge

After we’ve identified and defined the problem, the next step is to ask ourselves what we already know about the situation. Take an inventory. What information do we already have? What can we learn from the context? What resources have we been given?

When working with a student, pulling from existing knowledge might involve reviewing the concepts already covered and the student’s existing knowledge of the course material. It may also mean reaching into material and experiences outside of the student’s course.

Some helpful questions to guide this step include:

What does the student know about topics related to the course material?
What experience might the student have from prior courses?
In what context might the student have heard these ideas discussed in their everyday lives or in popular culture?

When we encourage students to step back and really take account of everything they already know about the problem and its context, they can be surprised at how much knowledge they actually bring.

Step 3: Refer to support materials

Once we’ve pulled from the knowledge we already have, we can expand our search for supporting knowledge to outside resources. Are there reference materials we can access? Are there experts we can consult?

The first thing we can encourage students to do is to refer to their course texts, notes, study guides, and materials provided by the class instructor. These are often the best places to start because they’re most likely to provide relevant information. Once these resources have been referenced, we can also encourage students to look for information and guidance from other academic resources.

Students often forget that they can reference what others have written about their problem. Outside textbooks and supporting texts may offer similar ideas presented in a different way, and this could help the student approach the problem with new understanding or perspective. Online research and reference materials are good places to look for clarification of rules, theories, laws, formulas, processes, and examples. While these sources may not be quite as specific to a student’s class assignment, they can sometimes provide confirmation or clarity in areas where a student might need it.

Students should be made to feel free to leverage other academic supports as well. They are already leveraging one aspect of this support when they come to see a tutor. Other supports may include making use of the library or computer center, visiting their instructor’s office hours to ask questions, or even reaching out to other classmates. It’s always helpful for tutors to remind the student that these other supports are available and to encourage them to use these resources.

If a student is unsure or intimidated by contacting an instructor or a classmate, or is uncomfortable learning how to use other support resources, encouragement from a tutor can often be the nudge a student needs. Remind them of these supports and offer to help them access them where appropriate.

Step 4: Brainstorm Solutions

There’s usually more than one way to solve a problem, and it’s helpful to brainstorm multiple solutions to find the one that works best.

It’s important that tutors allow students to take an active role in developing their own solutions to the problem. This is where our Socratic questioning skills become really crucial and can help the students to apply what they know to the problem they’ve identified. The tutor’s role here is to facilitate the solution-generating process, contributing where appropriate, and helping to guide the student in a productive direction.

It is possible that the student will suggest a solution that we know will not solve the problem. Depending on the nature and scale of the problem, it may not always be appropriate for us to tell the student that we think it won’t work. Guiding the student through the problem-solving process is about helping students to engage with the process itself. That way, they can feel confident applying it on their own, even when a helpful tutor isn’t around to give hints. It’s up to each tutor in each situation to decide when it is appropriate to expedite the process by providing insights into solutions, and when it is best to allow students to test their solutions to determine their effectiveness.

Step 5: Test a Solution

Choose a solution and try it out. Maybe it will work! Maybe it doesn’t. Having a variety of solutions to try is why we brainstorm more than one. Though trial and error can sometimes feel frustrating, it is in the testing of our solutions that we often learn the most. We’re able to better understand the parts that work, the parts that don’t, and hopefully learn the reasons why. This can result in solutions that are more efficient and better suited to our needs.

Solution-testing is an opportunity for students to learn from mistakes in a safe, low-risk way. Often mistakes in class result in deducted points, a bad grade, or maybe an embarrassing moment in from t of classmates. As a guide through the problem-solving process, tutors can help students to see mistakes as necessary and helpful steps on the way to a solution that works, rather than as failures. It’s important that the tutor help the student see mistakes as progress, especially when a student becomes discouraged. This helps the student maintain a growth mindset while identifying ways to improve.

Step 6: Revising the Solution

When a solution doesn’t work, it may not mean the whole idea was bad. Maybe it needs some revisions and refining, but doesn’t always need to be discarded. We can use what we learned from solution-testing to make effective revisions.

This may mean we guide a student back to previous steps in the problem-solving process. Students may once more need to pull from existing knowledge, revisit those support materials, or look at some of the alternative solutions that the student developed.

Step 7: Revisit the Problem

We’ve got a solution that works! Did it fix our problem? If yes, then great!

Sometimes, however, a solution may “work,” without fixing our problem.

When this happens, we need to revisit the problem definition. Do we really understand it? Is there a detail we didn’t consider when developing our solutions? Did we misinterpret what the problem actually is when we crafted our problem definition?

At this point, perhaps we need to revise the solution once more. Sometimes in our process of researching and brainstorming, we can get off course, and taking time to refer to the initial problem can help us recalibrate our efforts and get us back on track.

Other times we may need re-define our problem. Perhaps after developing and testing several solutions, it becomes clear that the real problem is different than what we initially thought it was. Or perhaps our solutions address parts of the problem, but don’t get to the deeper root of the issue.

When a student has worked through the problem-solving process and still feels stuck, tutors can guide them to revisit the problem and clarify the initial goal. Returning to previous steps of the process as needed is normal and often necessary. Ensuring students that they’re still correctly applying the process, even when they need to jump back and forth between these steps, can help keep them from getting discouraged.

Quickwrite Exercise

Think back to a time you solved a problem in the past. It could be an obstacle you encountered in an academic setting (completing an assignment, researching for a paper, troubleshooting a technical problem) or in your personal life.

Take a moment to reflect:

Did you use pieces of the rational problem solving process, without knowing?
If you could go back and approach the problem again, how would you implement this problem solving approach? What would it look like? How would it have been different?

Facilitating the Problem-Solving Process

The rational problem-solving process is an excellent tool to help tutors guide students through problems big and small. This organized way of approaching the task can help us make sure we’re heading in a productive direction, from solving a math problem to developing a strategy to finish a research paper. How do we ensure we’re empowering students to use this process on their own?

It can be helpful to both tutors and students to use the process as a checklist during a problem-solving session. We can name each step as we move through, and make it clear to the student the purpose of each activity. This doesn’t mean we turn a session of math tutoring into a lesson on the problem-solving process, but explicitly stating the names of each step can make it clear to the student the purpose of each activity, and help them to become familiar with the process. If we “narrate” our process as we go, students can experience a guided problem-solving process during their tutoring session and be encouraged to apply it independently.

Once we’ve guided a student through the process, we can then provide opportunities for the student to take charge. We can prompt the student to move from step to step, supporting them in their problem-solving efforts along the way. This guided practice can help students to become well-versed in the process itself, and to feel more comfortable applying it independently. 3

Something to Try

In your next session, when a student comes to you with a problem, use your Socratic questioning skills to walk the student through the problem solving process. (This may be something you’re already implementing naturally!)

Be deliberate about each step. Assist the student in defining the problem, guide the student to collect their existing knowledge, help the student pull from reference materials available, etc.

How does it work for you?

Practicing the Problem-Solving Process

Don’t forget, that while this process is an excellent tool for helping students to solve problems during a session, it can also help tutors to problem-solve during a session!

Perhaps you encounter a student faced with a problem you yourself don’t know how to solve. No worries! The problem-solving process works just the same.

We can apply it to challenges with assignments, and we can also apply it to other issues we encounter during a tutoring session. Every student is unique, and it may take some problem-solving to learn how to best work with each student. Identifying the “problem,” pulling from our knowledge, consulting our supports, brainstorming, and testing solutions are all ways tutors can determine how best to assist students.

Dane, E., Baer, M., Pratt, M. G., and Oldham, G. R. (2011). Rational versus intuitive problem solving: How thinking “off the beaten path” can stimulate creativity. Psychology of Aesthetics, Creativity, and the Arts. 5 (1), 3–12. https://doi.org/10.1037/a0017698.
Uzonwanne F.C. (2016). Rational Model of Decision Making. In: Farazmand A. (eds) Global Encyclopedia of Public Administration, Public Policy, and Governance. Springer, Cham. https://doi.org/10.1007/978-3-319-31816-5_2474-1.
Klegeris, A., Bahniwal, M., and Hurren, H. (2017). Improvement in Generic Problem-Solving Abilities of Students by Use of Tutor-less Problem-Based Learning in a Large Classroom Setting. Life Sciences Education. 12(1), 1-116. https://doi.org/10.1187/cbe.12-06-0081.

Additional Resources:

McNamera, C. (2020). Problem Solving and Decision Making (Solving Problems and Making Decisions). Free Management Library. Authenticity Consulting LLC. https://managementhelp.org/personalproductivity/problem-solving.htm . Accessed 26 Apr. 2021.

Nezu C., Palmatier, A., and Nezu, A. (2004). Social Problem-Solving Training for Caregivers. In Chang, D’Zurilla, & Sanna (Eds.) Social Problem Solving: Theory, Research, and Training. (223-238). American Psychological Association. https://doi.org/10.1037/10805-013 .

Nezu, A., Nezu, C., and D’Zurilla, T. (2007). Solving Life’s problems: a 5 Step Guide to Enhanced Well-Being. Springer Publishing Company LLC. https://www.springerpub.com/solving-life-s-problems-9780826114891.html .

Scott, G. M., Lonergan, D. C., and Mumford, M.D. (2010). Conceptual Combination: Alternative Knowledge Structures, Alternative Heuristics. Creativity Research Journal. 17(1), 79-98. https://www.tandfonline.com/doi/abs/10.1207/s15326934crj1701_7 .

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Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

Identify the Problem
Define the Problem
Form a Strategy
Organize Information
Allocate Resources
Monitor Progress
Evaluate the Results

Frequently Asked Questions

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 they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

Asking questions about the problem
Breaking the problem down into smaller pieces
Looking at the problem from different perspectives
Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

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You can become a better problem solving by:

Practicing brainstorming and coming up with multiple potential solutions to problems
Being open-minded and considering all possible options before making a decision
Breaking down problems into smaller, more manageable pieces
Asking for help when needed
Researching different problem-solving techniques and trying out new ones
Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors. The Psychology of Problem Solving . Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving . Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Our Mission

Using Mathematical Modeling to Get Real With Students

Unlike canned word problems, mathematical modeling plunges students into the messy complexities of real-world problem solving.

How do you bring math to life for kids? Illustrating the boundless possibilities of mathematics can be difficult if students are only asked to examine hypothetical situations like divvying up a dessert equally or determining how many apples are left after sharing with friends, writes third- and fourth- grade teacher Matthew Kandel for Mathematics Teacher: Learning and Teaching PK-12 .

In the early years of instruction, it’s not uncommon for students to think they’re learning math for the sole purpose of being able to solve word problems or help fictional characters troubleshoot issues in their imaginary lives, Kandel says. “A word problem is a one-dimensional world,” he writes. “Everything is distilled down to the quantities of interest. To solve a word problem, students can pick out the numbers and decide on an operation.”

But through the use of mathematical modeling, students are plucked out of the hypothetical realm and plunged into the complexities of reality—presented with opportunities to help solve real-world problems with many variables by generating questions, making assumptions, learning and applying new skills, and ultimately arriving at an answer.

In Kandel’s classroom, this work begins with breaking students into small groups, providing them with an unsharpened pencil and a simple, guiding question: “How many times can a pencil be sharpened before it is too small to use?”

Setting the Stage for Inquiry

The process of tackling the pencil question is not unlike the scientific method. After defining a question to investigate, students begin to wonder and hypothesize—what information do we need to know?—in order to identify a course of action. This step is unique to mathematical modeling: Whereas a word problem is formulaic, leading students down a pre-existing path toward a solution, a modeling task is “free-range,” empowering students to use their individual perspectives to guide them as they progress through their investigation, Kandel says.

Modeling problems also have a number of variables, and students themselves have the agency to determine what to ignore and what to focus their attention on.

After inter-group discussions, students in Kandel’s classroom came to the conclusion that they’d need answers to a host of other questions to proceed with answering their initial inquiry:

How much does the pencil sharpener remove?
What is the length of a brand new, unsharpened pencil?
Does the pencil sharpener remove the same amount of pencil each time it is used?

Introducing New Skills in Context

Once students have determined the first mathematical question they’d like to tackle (does the pencil sharpener remove the same amount of pencil each time it is used?), they are met with a roadblock. How were they to measure the pencil if the length did not fall conveniently on an inch or half inch? Kandel took the opportunity to introduce a new target skill which the class could begin using immediately: measuring to the nearest quarter inch.

“One group of students was not satisfied with the precision of measuring to the nearest quarter inch and asked to learn how to measure to the nearest eighth of an inch,” Kandel explains. “The attention and motivation exhibited by students is unrivaled by the traditional class in which the skill comes first, the problem second.”

Students reached a consensus and settled on taking six measurements total: the initial length of the new, unsharpened pencil as well as the lengths of the pencil after each of five sharpenings. To ensure all students can practice their newly acquired skill, Kandel tells the class that “all group members must share responsibility, taking turns measuring and checking the measurements of others.”

Next, each group created a simple chart to record their measurements, then plotted their data as a line graph—though exploring other data visualization techniques or engaging students in alternative followup activities would work as well.

“We paused for a quick lesson on the number line and the introduction of a new term—mixed numbers,” Kandel explains. “Armed with this new information, students had no trouble marking their y-axis in half- or quarter-inch increments.”

Sparking Mathematical Discussions

Mathematical modeling presents a multitude of opportunities for class-wide or small-group discussions, some which evolve into debates in which students state their hypotheses, then subsequently continue working to confirm or refute them.

Kandel’s students, for example, had a wide range of opinions when it came to answering the question of how small of a pencil would be deemed unusable. Eventually, the class agreed that once a pencil reached 1 ¼ inch, it could no longer be sharpened—though some students said they would be able to still write with it.

“This discussion helped us better understand what it means to make an assumption and how our assumptions affected our mathematical outcomes,” Kandel writes. Students then indicated the minimum size with a horizontal line across their respective graphs.

Many students independently recognized the final step of extending their line while looking at their graphs. With each of the six points representing their measurements, the points descended downward toward the newly added horizontal “line of inoperability.”

With mathematical modeling, Kandel says, there are no right answers, only models that are “more or less closely aligned with real-world observations.” Each group of students may come to a different conclusion, which can lead to a larger class discussion about accuracy. To prove their group had the most accurate conclusion, students needed to compare and contrast their methods as well as defend their final result.

Developing Your Own Mathematical Models

The pencil problem is a great starting point for introducing mathematical modeling and free-range problem solving to your students, but you can customize based on what you have available and the particular needs of each group of students.

Depending on the type of pencil sharpener you have, for example, students can determine what constitutes a “fair test” and set the terms of their own inquiry.

Additionally, Kandel suggests putting scaffolds in place to allow students who are struggling with certain elements to participate: Simplified rulers can be provided for students who need accommodations; charts can be provided for students who struggle with data collection; graphs with prelabeled x- and y-axes can be prepared in advance.

.css-1sk4066:hover{background:#d1ecfa;} 7 Real-World Math Strategies

Students can also explore completely different free-range problem solving and real world applications for math . At North Agincourt Jr. Public School in Scarborough, Canada, kids in grades 1-6 learn to conduct water audits. By adding, subtracting, finding averages, and measuring liquids—like the flow rate of all the water foundations, toilets, and urinals—students measure the amount of water used in their school or home in a single day.

Or you can ask older students to bring in common household items—anything from a measuring cup to a recipe card—and identify three ways the item relates to math. At Woodrow Petty Elementary School in Taft, Texas, fifth-grade students display their chosen objects on the class’s “real-world math wall.” Even acting out restaurant scenarios can provide students with an opportunity to reinforce critical mathematical skills like addition and subtraction, while bolstering an understanding of decimals and percentages. At Suzhou Singapore International School in China, third- to fifth- graders role play with menus, ordering fictional meals and learning how to split the check when the bill arrives.

How to master the seven-step problem-solving process

In this episode of the McKinsey Podcast , Simon London speaks with Charles Conn, CEO of venture-capital firm Oxford Sciences Innovation, and McKinsey senior partner Hugo Sarrazin about the complexities of different problem-solving strategies.

Podcast transcript

Simon London: Hello, and welcome to this episode of the McKinsey Podcast , with me, Simon London. What’s the number-one skill you need to succeed professionally? Salesmanship, perhaps? Or a facility with statistics? Or maybe the ability to communicate crisply and clearly? Many would argue that at the very top of the list comes problem solving: that is, the ability to think through and come up with an optimal course of action to address any complex challenge—in business, in public policy, or indeed in life.

Looked at this way, it’s no surprise that McKinsey takes problem solving very seriously, testing for it during the recruiting process and then honing it, in McKinsey consultants, through immersion in a structured seven-step method. To discuss the art of problem solving, I sat down in California with McKinsey senior partner Hugo Sarrazin and also with Charles Conn. Charles is a former McKinsey partner, entrepreneur, executive, and coauthor of the book Bulletproof Problem Solving: The One Skill That Changes Everything [John Wiley & Sons, 2018].

Charles and Hugo, welcome to the podcast. Thank you for being here.

Hugo Sarrazin: Our pleasure.

Charles Conn: It’s terrific to be here.

Simon London: Problem solving is a really interesting piece of terminology. It could mean so many different things. I have a son who’s a teenage climber. They talk about solving problems. Climbing is problem solving. Charles, when you talk about problem solving, what are you talking about?

Charles Conn: For me, problem solving is the answer to the question “What should I do?” It’s interesting when there’s uncertainty and complexity, and when it’s meaningful because there are consequences. Your son’s climbing is a perfect example. There are consequences, and it’s complicated, and there’s uncertainty—can he make that grab? I think we can apply that same frame almost at any level. You can think about questions like “What town would I like to live in?” or “Should I put solar panels on my roof?”

You might think that’s a funny thing to apply problem solving to, but in my mind it’s not fundamentally different from business problem solving, which answers the question “What should my strategy be?” Or problem solving at the policy level: “How do we combat climate change?” “Should I support the local school bond?” I think these are all part and parcel of the same type of question, “What should I do?”

I’m a big fan of structured problem solving. By following steps, we can more clearly understand what problem it is we’re solving, what are the components of the problem that we’re solving, which components are the most important ones for us to pay attention to, which analytic techniques we should apply to those, and how we can synthesize what we’ve learned back into a compelling story. That’s all it is, at its heart.

I think sometimes when people think about seven steps, they assume that there’s a rigidity to this. That’s not it at all. It’s actually to give you the scope for creativity, which often doesn’t exist when your problem solving is muddled.

Simon London: You were just talking about the seven-step process. That’s what’s written down in the book, but it’s a very McKinsey process as well. Without getting too deep into the weeds, let’s go through the steps, one by one. You were just talking about problem definition as being a particularly important thing to get right first. That’s the first step. Hugo, tell us about that.

Hugo Sarrazin: It is surprising how often people jump past this step and make a bunch of assumptions. The most powerful thing is to step back and ask the basic questions—“What are we trying to solve? What are the constraints that exist? What are the dependencies?” Let’s make those explicit and really push the thinking and defining. At McKinsey, we spend an enormous amount of time in writing that little statement, and the statement, if you’re a logic purist, is great. You debate. “Is it an ‘or’? Is it an ‘and’? What’s the action verb?” Because all these specific words help you get to the heart of what matters.

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Simon London: So this is a concise problem statement.

Hugo Sarrazin: Yeah. It’s not like “Can we grow in Japan?” That’s interesting, but it is “What, specifically, are we trying to uncover in the growth of a product in Japan? Or a segment in Japan? Or a channel in Japan?” When you spend an enormous amount of time, in the first meeting of the different stakeholders, debating this and having different people put forward what they think the problem definition is, you realize that people have completely different views of why they’re here. That, to me, is the most important step.

Charles Conn: I would agree with that. For me, the problem context is critical. When we understand “What are the forces acting upon your decision maker? How quickly is the answer needed? With what precision is the answer needed? Are there areas that are off limits or areas where we would particularly like to find our solution? Is the decision maker open to exploring other areas?” then you not only become more efficient, and move toward what we call the critical path in problem solving, but you also make it so much more likely that you’re not going to waste your time or your decision maker’s time.

How often do especially bright young people run off with half of the idea about what the problem is and start collecting data and start building models—only to discover that they’ve really gone off half-cocked.

Hugo Sarrazin: Yeah.

Charles Conn: And in the wrong direction.

Simon London: OK. So step one—and there is a real art and a structure to it—is define the problem. Step two, Charles?

Charles Conn: My favorite step is step two, which is to use logic trees to disaggregate the problem. Every problem we’re solving has some complexity and some uncertainty in it. The only way that we can really get our team working on the problem is to take the problem apart into logical pieces.

What we find, of course, is that the way to disaggregate the problem often gives you an insight into the answer to the problem quite quickly. I love to do two or three different cuts at it, each one giving a bit of a different insight into what might be going wrong. By doing sensible disaggregations, using logic trees, we can figure out which parts of the problem we should be looking at, and we can assign those different parts to team members.

Simon London: What’s a good example of a logic tree on a sort of ratable problem?

Charles Conn: Maybe the easiest one is the classic profit tree. Almost in every business that I would take a look at, I would start with a profit or return-on-assets tree. In its simplest form, you have the components of revenue, which are price and quantity, and the components of cost, which are cost and quantity. Each of those can be broken out. Cost can be broken into variable cost and fixed cost. The components of price can be broken into what your pricing scheme is. That simple tree often provides insight into what’s going on in a business or what the difference is between that business and the competitors.

If we add the leg, which is “What’s the asset base or investment element?”—so profit divided by assets—then we can ask the question “Is the business using its investments sensibly?” whether that’s in stores or in manufacturing or in transportation assets. I hope we can see just how simple this is, even though we’re describing it in words.

When I went to work with Gordon Moore at the Moore Foundation, the problem that he asked us to look at was “How can we save Pacific salmon?” Now, that sounds like an impossible question, but it was amenable to precisely the same type of disaggregation and allowed us to organize what became a 15-year effort to improve the likelihood of good outcomes for Pacific salmon.

Simon London: Now, is there a danger that your logic tree can be impossibly large? This, I think, brings us onto the third step in the process, which is that you have to prioritize.

Charles Conn: Absolutely. The third step, which we also emphasize, along with good problem definition, is rigorous prioritization—we ask the questions “How important is this lever or this branch of the tree in the overall outcome that we seek to achieve? How much can I move that lever?” Obviously, we try and focus our efforts on ones that have a big impact on the problem and the ones that we have the ability to change. With salmon, ocean conditions turned out to be a big lever, but not one that we could adjust. We focused our attention on fish habitats and fish-harvesting practices, which were big levers that we could affect.

People spend a lot of time arguing about branches that are either not important or that none of us can change. We see it in the public square. When we deal with questions at the policy level—“Should you support the death penalty?” “How do we affect climate change?” “How can we uncover the causes and address homelessness?”—it’s even more important that we’re focusing on levers that are big and movable.

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Simon London: Let’s move swiftly on to step four. You’ve defined your problem, you disaggregate it, you prioritize where you want to analyze—what you want to really look at hard. Then you got to the work plan. Now, what does that mean in practice?

Hugo Sarrazin: Depending on what you’ve prioritized, there are many things you could do. It could be breaking the work among the team members so that people have a clear piece of the work to do. It could be defining the specific analyses that need to get done and executed, and being clear on time lines. There’s always a level-one answer, there’s a level-two answer, there’s a level-three answer. Without being too flippant, I can solve any problem during a good dinner with wine. It won’t have a whole lot of backing.

Simon London: Not going to have a lot of depth to it.

Hugo Sarrazin: No, but it may be useful as a starting point. If the stakes are not that high, that could be OK. If it’s really high stakes, you may need level three and have the whole model validated in three different ways. You need to find a work plan that reflects the level of precision, the time frame you have, and the stakeholders you need to bring along in the exercise.

Charles Conn: I love the way you’ve described that, because, again, some people think of problem solving as a linear thing, but of course what’s critical is that it’s iterative. As you say, you can solve the problem in one day or even one hour.

Charles Conn: We encourage our teams everywhere to do that. We call it the one-day answer or the one-hour answer. In work planning, we’re always iterating. Every time you see a 50-page work plan that stretches out to three months, you know it’s wrong. It will be outmoded very quickly by that learning process that you described. Iterative problem solving is a critical part of this. Sometimes, people think work planning sounds dull, but it isn’t. It’s how we know what’s expected of us and when we need to deliver it and how we’re progressing toward the answer. It’s also the place where we can deal with biases. Bias is a feature of every human decision-making process. If we design our team interactions intelligently, we can avoid the worst sort of biases.

Simon London: Here we’re talking about cognitive biases primarily, right? It’s not that I’m biased against you because of your accent or something. These are the cognitive biases that behavioral sciences have shown we all carry around, things like anchoring, overoptimism—these kinds of things.

Both: Yeah.

Charles Conn: Availability bias is the one that I’m always alert to. You think you’ve seen the problem before, and therefore what’s available is your previous conception of it—and we have to be most careful about that. In any human setting, we also have to be careful about biases that are based on hierarchies, sometimes called sunflower bias. I’m sure, Hugo, with your teams, you make sure that the youngest team members speak first. Not the oldest team members, because it’s easy for people to look at who’s senior and alter their own creative approaches.

Hugo Sarrazin: It’s helpful, at that moment—if someone is asserting a point of view—to ask the question “This was true in what context?” You’re trying to apply something that worked in one context to a different one. That can be deadly if the context has changed, and that’s why organizations struggle to change. You promote all these people because they did something that worked well in the past, and then there’s a disruption in the industry, and they keep doing what got them promoted even though the context has changed.

Simon London: Right. Right.

Hugo Sarrazin: So it’s the same thing in problem solving.

Charles Conn: And it’s why diversity in our teams is so important. It’s one of the best things about the world that we’re in now. We’re likely to have people from different socioeconomic, ethnic, and national backgrounds, each of whom sees problems from a slightly different perspective. It is therefore much more likely that the team will uncover a truly creative and clever approach to problem solving.

Simon London: Let’s move on to step five. You’ve done your work plan. Now you’ve actually got to do the analysis. The thing that strikes me here is that the range of tools that we have at our disposal now, of course, is just huge, particularly with advances in computation, advanced analytics. There’s so many things that you can apply here. Just talk about the analysis stage. How do you pick the right tools?

Charles Conn: For me, the most important thing is that we start with simple heuristics and explanatory statistics before we go off and use the big-gun tools. We need to understand the shape and scope of our problem before we start applying these massive and complex analytical approaches.

Simon London: Would you agree with that?

Hugo Sarrazin: I agree. I think there are so many wonderful heuristics. You need to start there before you go deep into the modeling exercise. There’s an interesting dynamic that’s happening, though. In some cases, for some types of problems, it is even better to set yourself up to maximize your learning. Your problem-solving methodology is test and learn, test and learn, test and learn, and iterate. That is a heuristic in itself, the A/B testing that is used in many parts of the world. So that’s a problem-solving methodology. It’s nothing different. It just uses technology and feedback loops in a fast way. The other one is exploratory data analysis. When you’re dealing with a large-scale problem, and there’s so much data, I can get to the heuristics that Charles was talking about through very clever visualization of data.

You test with your data. You need to set up an environment to do so, but don’t get caught up in neural-network modeling immediately. You’re testing, you’re checking—“Is the data right? Is it sound? Does it make sense?”—before you launch too far.

Simon London: You do hear these ideas—that if you have a big enough data set and enough algorithms, they’re going to find things that you just wouldn’t have spotted, find solutions that maybe you wouldn’t have thought of. Does machine learning sort of revolutionize the problem-solving process? Or are these actually just other tools in the toolbox for structured problem solving?

Charles Conn: It can be revolutionary. There are some areas in which the pattern recognition of large data sets and good algorithms can help us see things that we otherwise couldn’t see. But I do think it’s terribly important we don’t think that this particular technique is a substitute for superb problem solving, starting with good problem definition. Many people use machine learning without understanding algorithms that themselves can have biases built into them. Just as 20 years ago, when we were doing statistical analysis, we knew that we needed good model definition, we still need a good understanding of our algorithms and really good problem definition before we launch off into big data sets and unknown algorithms.

Simon London: Step six. You’ve done your analysis.

Charles Conn: I take six and seven together, and this is the place where young problem solvers often make a mistake. They’ve got their analysis, and they assume that’s the answer, and of course it isn’t the answer. The ability to synthesize the pieces that came out of the analysis and begin to weave those into a story that helps people answer the question “What should I do?” This is back to where we started. If we can’t synthesize, and we can’t tell a story, then our decision maker can’t find the answer to “What should I do?”

Simon London: But, again, these final steps are about motivating people to action, right?

Charles Conn: Yeah.

Simon London: I am slightly torn about the nomenclature of problem solving because it’s on paper, right? Until you motivate people to action, you actually haven’t solved anything.

Charles Conn: I love this question because I think decision-making theory, without a bias to action, is a waste of time. Everything in how I approach this is to help people take action that makes the world better.

Simon London: Hence, these are absolutely critical steps. If you don’t do this well, you’ve just got a bunch of analysis.

Charles Conn: We end up in exactly the same place where we started, which is people speaking across each other, past each other in the public square, rather than actually working together, shoulder to shoulder, to crack these important problems.

Simon London: In the real world, we have a lot of uncertainty—arguably, increasing uncertainty. How do good problem solvers deal with that?

Hugo Sarrazin: At every step of the process. In the problem definition, when you’re defining the context, you need to understand those sources of uncertainty and whether they’re important or not important. It becomes important in the definition of the tree.

You need to think carefully about the branches of the tree that are more certain and less certain as you define them. They don’t have equal weight just because they’ve got equal space on the page. Then, when you’re prioritizing, your prioritization approach may put more emphasis on things that have low probability but huge impact—or, vice versa, may put a lot of priority on things that are very likely and, hopefully, have a reasonable impact. You can introduce that along the way. When you come back to the synthesis, you just need to be nuanced about what you’re understanding, the likelihood.

Often, people lack humility in the way they make their recommendations: “This is the answer.” They’re very precise, and I think we would all be well-served to say, “This is a likely answer under the following sets of conditions” and then make the level of uncertainty clearer, if that is appropriate. It doesn’t mean you’re always in the gray zone; it doesn’t mean you don’t have a point of view. It just means that you can be explicit about the certainty of your answer when you make that recommendation.

Simon London: So it sounds like there is an underlying principle: “Acknowledge and embrace the uncertainty. Don’t pretend that it isn’t there. Be very clear about what the uncertainties are up front, and then build that into every step of the process.”

Hugo Sarrazin: Every step of the process.

Simon London: Yeah. We have just walked through a particular structured methodology for problem solving. But, of course, this is not the only structured methodology for problem solving. One that is also very well-known is design thinking, which comes at things very differently. So, Hugo, I know you have worked with a lot of designers. Just give us a very quick summary. Design thinking—what is it, and how does it relate?

Hugo Sarrazin: It starts with an incredible amount of empathy for the user and uses that to define the problem. It does pause and go out in the wild and spend an enormous amount of time seeing how people interact with objects, seeing the experience they’re getting, seeing the pain points or joy—and uses that to infer and define the problem.

Simon London: Problem definition, but out in the world.

Hugo Sarrazin: With an enormous amount of empathy. There’s a huge emphasis on empathy. Traditional, more classic problem solving is you define the problem based on an understanding of the situation. This one almost presupposes that we don’t know the problem until we go see it. The second thing is you need to come up with multiple scenarios or answers or ideas or concepts, and there’s a lot of divergent thinking initially. That’s slightly different, versus the prioritization, but not for long. Eventually, you need to kind of say, “OK, I’m going to converge again.” Then you go and you bring things back to the customer and get feedback and iterate. Then you rinse and repeat, rinse and repeat. There’s a lot of tactile building, along the way, of prototypes and things like that. It’s very iterative.

Simon London: So, Charles, are these complements or are these alternatives?

Charles Conn: I think they’re entirely complementary, and I think Hugo’s description is perfect. When we do problem definition well in classic problem solving, we are demonstrating the kind of empathy, at the very beginning of our problem, that design thinking asks us to approach. When we ideate—and that’s very similar to the disaggregation, prioritization, and work-planning steps—we do precisely the same thing, and often we use contrasting teams, so that we do have divergent thinking. The best teams allow divergent thinking to bump them off whatever their initial biases in problem solving are. For me, design thinking gives us a constant reminder of creativity, empathy, and the tactile nature of problem solving, but it’s absolutely complementary, not alternative.

Simon London: I think, in a world of cross-functional teams, an interesting question is do people with design-thinking backgrounds really work well together with classical problem solvers? How do you make that chemistry happen?

Hugo Sarrazin: Yeah, it is not easy when people have spent an enormous amount of time seeped in design thinking or user-centric design, whichever word you want to use. If the person who’s applying classic problem-solving methodology is very rigid and mechanical in the way they’re doing it, there could be an enormous amount of tension. If there’s not clarity in the role and not clarity in the process, I think having the two together can be, sometimes, problematic.

The second thing that happens often is that the artifacts the two methodologies try to gravitate toward can be different. Classic problem solving often gravitates toward a model; design thinking migrates toward a prototype. Rather than writing a big deck with all my supporting evidence, they’ll bring an example, a thing, and that feels different. Then you spend your time differently to achieve those two end products, so that’s another source of friction.

Now, I still think it can be an incredibly powerful thing to have the two—if there are the right people with the right mind-set, if there is a team that is explicit about the roles, if we’re clear about the kind of outcomes we are attempting to bring forward. There’s an enormous amount of collaborativeness and respect.

Simon London: But they have to respect each other’s methodology and be prepared to flex, maybe, a little bit, in how this process is going to work.

Hugo Sarrazin: Absolutely.

Simon London: The other area where, it strikes me, there could be a little bit of a different sort of friction is this whole concept of the day-one answer, which is what we were just talking about in classical problem solving. Now, you know that this is probably not going to be your final answer, but that’s how you begin to structure the problem. Whereas I would imagine your design thinkers—no, they’re going off to do their ethnographic research and get out into the field, potentially for a long time, before they come back with at least an initial hypothesis.

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Hugo Sarrazin: That is a great callout, and that’s another difference. Designers typically will like to soak into the situation and avoid converging too quickly. There’s optionality and exploring different options. There’s a strong belief that keeps the solution space wide enough that you can come up with more radical ideas. If there’s a large design team or many designers on the team, and you come on Friday and say, “What’s our week-one answer?” they’re going to struggle. They’re not going to be comfortable, naturally, to give that answer. It doesn’t mean they don’t have an answer; it’s just not where they are in their thinking process.

Simon London: I think we are, sadly, out of time for today. But Charles and Hugo, thank you so much.

Charles Conn: It was a pleasure to be here, Simon.

Hugo Sarrazin: It was a pleasure. Thank you.

Simon London: And thanks, as always, to you, our listeners, for tuning into this episode of the McKinsey Podcast . If you want to learn more about problem solving, you can find the book, Bulletproof Problem Solving: The One Skill That Changes Everything , online or order it through your local bookstore. To learn more about McKinsey, you can of course find us at McKinsey.com.

Charles Conn is CEO of Oxford Sciences Innovation and an alumnus of McKinsey’s Sydney office. Hugo Sarrazin is a senior partner in the Silicon Valley office, where Simon London, a member of McKinsey Publishing, is also based.

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Strategy to beat the odds

Five routes to more innovative problem solving

17 Problem Solving & Modeling

Introduction.

Aerospace engineering students soon begin to ask when they can start to solve actual problems relevant to aircraft, rockets, or spacecraft. Of course, it is natural to ask such questions, even early on in an engineering education. However, practical problem-solving in engineering is a serious business that requires engineers-in-training to become well-versed in the fundamental subjects appropriate to their field. Besides the usual engineering disciplines, this process of learning and becoming proficient will inevitably include having a solid background in physics, chemistry, mathematics, numerical methods, and computer programming.

In general, engineering problem-solving is an established and well-proven process in which the bigger problem is first dissected into smaller, more manageable, and perhaps better digestible parts, as illustrated in the figure below for a flight vehicle. Then, each vehicle component can be analyzed separately, at least initially, perhaps with experiments, mathematical models, numerical models, or all these working approaches working hand-in-hand. Finally, the understanding and functionality of the parts can be reassembled using a synthesis approach to understand the flight vehicle as a whole, with interactions considered, too. Of course, such an approach is always flawed, but it forms at least one rational basis for understanding and designing complex engineering systems, including aircraft and spacecraft.

Today, aerospace engineers must also have increasingly multidisciplinary technical skill sets, which means they must follow a broader-based educational path and become more knowledgeable and versatile in using a more comprehensive range of subject matter. It is no longer sufficient to be an aerodynamics specialist or “aerodynamicist,” or a “structural dynamicist” or an “acoustician.” Successful aerospace engineers must be technically broad and increasingly versatile to work with others in interdisciplinary contexts.

Furthermore, using artificial intelligence (or AI), machine learning, and big data analytics is increasingly prevalent in the aerospace industry, for which engineers must know the benefits and limitations. As the industry evolves and solves more challenging problems, it is also crucial that aerospace engineers continue to develop their problem-solving skills throughout their careers to stay current with the latest technological advancements.

Learning Objectives

Begin to understand the fundamental processes involved in engineering problem-solving.
Appreciate the significance of governing equations and how such equations can be reduced so that they apply to specific problems.
Understand the ideas of modeling complexity and the trades between fidelity and cost.
Better understand the importance and expectations of doing homework problems in engineering classes.

Hypothetico-Deductive Method

Many have argued that the engineering design process must follow the hypothetico-deductive method (or H-D method), a primary method for testing hypotheses or conjectures. The “hypothetico” (or hypothetical) part is where a hypothesis or theory is proposed, which needs to be tested, and the “deductive” part is where the consequences are drawn from the hypothesis or hypotheses. [1] The H-D method is sometimes called THE scientific method , but it is not the only method used in scientific and engineering work. The H-D method can be divided into the four stages outlined in the figure below.

1. Identify the theory, the hypothesis, or the conjecture to be tested . This approach does not necessarily need to rely on facts and allows for “imaginative preconceptions, intuition, and even luck.” However, the hypothesis usually relies on prior understanding or awareness of pervasive laws.

2. Generate predictions from the theory. The theories are used to make predictions about what we see, i.e., we proceed to imitate what we perceive as the “real world.” These predictions would also encompass the range of conditions perceived as the theory’s domain of applicability, which cannot necessarily have limitless bounds and will trade cost with theoretical complexity and value with predictive fidelity.

3. Use various types of experiments and measurements to test whether predictions are, in fact, correct. If done correctly, the measurements represent the truth. The data acquired, assuming the quantities needed can be measured to the necessary levels of fidelity, provides the evidence to test the proposed theories. Replication of an experiment by others and repeatability of the data over time are critical to doing good science. The data may often uncover unexpected outcomes and spawn new directions if sufficiently comprehensive and high-quality.

4. Expose the theory to criticism, then reject or modify the theory or declare that the theory has been validated or otherwise proved. This process may take significant time and often depends on specific experimental measurements and other data availability. However, it is the most crucial part of doing scientific research. Alternative hypotheses may be pursued after criticism to see which is more likely to explain the predictions. In this regard, the principle of parsimony, or Ockham’s Razor , is essential in mathematical modeling, i.e., in generating equations and mathematical models to represent any given physical behavior.

The goal is to keep modeling complexity and predictive fidelity in equilibrium. This can be achieved through careful, systematic validation studies and requires a degree of engineering common sense. After all, as the figure below suggests, this is the ultimate foundation on which the scientific method is based.

Other methods are used in scientific and engineering work. For example, descriptive methods observe and describe phenomena, while experimental methods manipulate variables to establish causal relationships. Correlational studies examine the relationships between variables without manipulation. Qualitative research gathers in-depth insights through non-numerical data, while mixed-methods research combines quantitative and qualitative approaches. Longitudinal studies track changes over time, meta-analysis synthesizes findings from multiple studies, and action research collaboratively addresses practical issues. Each method offers unique strengths suited to different research questions and objectives, guiding scientific exploration across diverse disciplines and contexts. Nevertheless, the hypothetico-deductive method remains the primary method for testing scientific and engineering hypotheses.

Starting Out

As students think more about engineering concepts and various types of problem-solving in aerodynamics, structures, flight vehicle performance, and other areas, some words of caution are appropriate. First, it must be appreciated that there are few “handy equations” for solving engineering problems, especially in aerodynamics. Instead, the relevant equations for problem-solving must be selected carefully in terms of the specific equations that most appropriately govern the problem, called the governing equations . Choosing the governing equations is one issue, but solving them using the correct boundary conditions, and perhaps with any appropriate simplifications, involves considerable skill that comes only from much practice, i.e., consistent, purposeful, and focused practice over time. The solutions to some of these problems become homework exemplars of the field. “ Those who study a scientific discipline are expected to know its exemplars .” There is no fixed set of exemplars in aerospace engineering; however, this ebook is filled with a few hundred for starters.

“When you can measure what you are speaking about and express it in numbers, you know something about it; but when you cannot measure it when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science , whatever the matter may be.” Source: S ir William Thompson (Lord Kevin), Popular Lectures and Addresses, V ol. 1 (1889) “Electrical Units of Measurement,” from a presentation delivered on May 3, 1883.

Aerodynamics

Remember that aerodynamics is the underpinning of flight, so some form of aerodynamic analysis, such as shown in the figure below, comes into almost all types of problem-solving with aircraft and even with spacecraft, i.e., launch and re-entry vehicles. Therefore, it is essential that the selected aerodynamic models be sufficiently comprehensive and detailed to predict what is needed and for the right reason. Ultimately, they will provide a more substantial basis for decision-making, resource allocation, risk management, and adaptability. By considering a wide range of factors and understanding the underlying reasons, it becomes possible to navigate options and reduce uncertainties to make informed choices.

Value-Based Analysis

How comprehensive and detailed any mathematical model needs to be will depend on the specific type and complexity of the physical problem, which also affects the time and effort (i.e., cost) to solve the problem. If the model is intended for precise predictions, design optimization, or critical engineering applications, a higher level of detail and accuracy is typically required, and the penalty is time for a solution. In the workplace, time equates to money, i.e., a value-based decision process is needed, so it is essential to balance the technical level of detail in the model and the resulting computational cost.

Mathematical models must consider additional factors such as non-linearities, coupling effects, boundary interactions, or time-dependent behavior. These complexities demand more comprehensive models that incorporate more detail to represent the physical problem and its behavior accurately at the price of longer execution time and higher costs. For example, aerodynamic equations may need to be solved consistently and simultaneously with other sets of equations describing the behavior of different aspects of the aircraft, such as its dynamic flight motion, aeroelasticity, acoustics, etc.

It is soon concluded that problem-solving in engineering, especially concerning flight vehicles, can be very challenging, time-consuming, and quite costly because the forms of the governing equations for different disciplines will inevitably be different. This issue means different solution methods, including the various specific techniques, numerical methods, etc., may be required in each case, e.g., numerically solving together or coupling the governing equations that are applicable and traditionally used in the different engineering fields.

Governing Equations

As students learn more about aerospace engineering disciplines, they will be exposed to more general forms of potentially applicable governing equations in each field. In many cases, the relevant governing equations that apply to a given problem may be subsets of more general governing equations intended to apply to a broader range of problems and conditions. In other cases, the governing equations may need to be developed from first principles, e.g., directly invoking the conservation principles of mechanics and thermodynamics. However, the approach must be systematic when choosing or creating the equations that apply to or govern a specific problem. The approach of picking a few equations and hoping they will apply is called an ad hoc approach. However, such an approach will inevitably prove disastrous; engineering case histories and experience speak for themselves.

Problem-solving also requires that the meaning of the governing equations must be understood, as well as each of the terms that comprise these equations. There may be terms or groups of terms in these equations that have different levels of complexity and then may also have interdependencies, so dropping one term may have unintended consequences on the evaluation of other terms. In many cases, simplified (reduced) or otherwise particular governing equations may be appropriate because they simplify the solution process. Indeed, an approximate (and fast) solution might be adequate in the initial design phases. However, a more accurate (and likely more time-consuming) solution might be needed later. Part of the skill that must be developed in engineering problem-solving is to decide what terms in the equations must be retained and what terms can be eliminated without substantially affecting the outcomes of the final solution.

Setting Up Aerodynamic Models

Solving problems in aerodynamics requires that appropriate mathematical models of the flow are set up correctly. The derivation of the mathematical equations that describe fluid dynamic and aerodynamic flows is a systematic process that has become well-established in engineering practice. However, because air (like all fluids) will continuously deform as it flows, its behavior must naturally be expected to be more difficult to describe than a solid material.

The starting point of any aerodynamic analysis is the statement of the physical problem, a definition of the appropriate boundary conditions, and making justifiable assumptions and approximations about how the flow develops. An example of a boundary condition is one that defines the values of the free-stream flow conditions or how the flow behaves on a boundary or surface, e.g., it flows parallel to the surface. The three fundamental conservation principles of mechanics must then be applied to the aerodynamic problem, namely:

Conservation of mass, i.e., mass is neither created nor destroyed.
Conservation of momentum, i.e., a force applied to a mass equals its time rate of change of momentum.
Conservation of energy, i.e., energy is neither created nor destroyed and can only be converted from one form into another.

The resulting mathematical equations should then describe the aerodynamic behavior of the flow of interest, at least within the bounds of the stated assumptions and approximations. The solution to these equations can proceed analytically, numerically, or both, hopefully giving an engineer the desired results.

All practical problems will inevitably require some assumptions and approximations to obtain solutions. A common assumption is that air behaves as an ideal gas. Other assumptions might include two-dimensional flow, steady flow, incompressible flow, inviscid flow, etc. However, all such assumptions must be justified, and such justifications often take skill and experience. Skill and experience are obtained through solving engineering problems, which first develops from diligently doing homework problems.

For example, the figure below shows a hierarchy of aerodynamic methods that could be used to model the real flow, starting from lifting line theory, passing through lifting surface theory to a panel method that also models the effects of wing thickness, and finally to a full computational fluid dynamics (CFD) model. Each method is accompanied by a commensurate increase in fidelity and an increase in execution time and computational cost. Whether the problem is aerodynamics or otherwise, the idea is often to start with a more straightforward method to get some initial understanding of the problem at a relatively low cost and then progress to a more complex model with higher fidelity for the final calculations. For example, a CFD calculation may take up to five orders of magnitude more time and cost than the lifting line theory.

Verification and Validation

Inevitably, the outcomes from any model (i.e., the computed results) would be checked to make sure that they make sense, such as by comparing them with measurements, i.e., to establish how well the model works, a process called verification and validation or “V & V.” If the outcomes are positive, then the process inevitably also involves checking it for different input scenarios to ensure that the model predicts what it needs to predict and for the right reason. However, the process may conclude that the results do not agree. The V & V of all mathematical models is critically important, especially if such models are to be used in the design process. Confidence in the design tools being used is critical if the design is to prove viable or successful.

An example is shown in the figure below, where predictions from four mathematical models (i.e., different competing methods or “theories”) are compared against measurements. The question is: “Which mathematical model works the best compared to measurements?” All methods work reasonably well over some ranges, but one or more may be better than the others. The exact theory fails for higher values of the independent variable. Subjectively, Method 3 seems to compare best to the measurement points. However, when measurement uncertainty is accounted for, Methods 1 and 2 may be just as good regarding their predicted values. Unfortunately, in this case, the final answer as to the “best” model may be in the eye of the beholder! The other issue to examine is to ensure that the best method predicts the behavior for the right reason, which may take additional measurements and/or analysis to answer fully. Predicting the correct outcome for the wrong reason is not a verification or validation of the method. Indeed, some may argue that such models cannot be validated, only disproved, i.e., the essence of Popper’s falsification principle.

Finally, these equations (i.e., the “model,” in general) can be used to analyze similar problems of interest, such as in a parametric study where one quantity is varied using the model to establish the corresponding effects on the other quantities. Such parametric studies with mathematical and/or numerical models are critical in engineering design. For example, a particular combination of parameters may be sought to optimize the system performance or for another reason. However, it should always be remembered that no mathematical description of a physical problem can be perfect for its behavior. It can only be an approximation whose accuracy depends on how diligently the model is set up, including the nature of the assumptions and approximations.

Multi-Disciplinary Problems

As previously discussed, aircraft and spacecraft structures are lightweight, thin-walled structures of various beams, columns, shafts, plates, shells, etc. All must be modeled in aggregate with aerodynamics using a multi-disciplinary approach, the idea being shown in the figure below. Coupling aerodynamics with the structures and structural dynamics is called aeroelasticity because the action of the aerodynamic loads will elastically deform the structure and so will feed back to the aerodynamic loads. Combining models of aerodynamics and structural loads and deformations usually requires an iterative approach, adding significantly to the computational time and overall effort.

Like aerodynamics, the governing equations of structural elasticity comprise sets of partial differential equations. Computational Fluid Dynamics (CFD) can help predict the detailed aerodynamic loads if sufficient computing power is available, i.e., the need for memory and execution speed. For the structure, the finite element method (FEM) is usually necessary to predict the behavior of complex structural geometries. The FEM is highly developed and sophisticated enough to handle just about any aerospace structure, but like CFD also requires significant computing power. Sometimes, subsets of the governing equations may be adequate to speed up the computational process, depending on the assumptions and the approximations that can be justified, e.g., small displacements or isotropic structural properties.

Flutter is a form of aeroelasticity and is a potentially catastrophic dynamic phenomenon that can happen with the inherently flexible structures of aircraft and spacecraft. Flutter usually occurs when the forces created on an object cause it to displace or deform, elastically return to where it was, but also overshoot and then repeat the process and begin to oscillate, i.e., it is a dynamic process requiring an inherent coupled aerodynamic and structural dynamic solution process. If the forces subsequently increase in magnitude, the oscillations also increase until the object eventually fails structurally, e.g., the tail or a wing may break off during flight. Therefore, flutter must be avoided, and great efforts must be undertaken to ensure a flight vehicle is flutter-free. Unfortunately, flutter can still occur on flight vehicles, and it can also happen with buildings, bridges, and other flexible objects exposed to the wind.

Problem-Solving Process

The figure below indicates that a general procedure for solving a physical problem requires many steps. The steps are not unique and will vary from problem to problem, but the iterative process is typical to all aspects of design, particularly for flight vehicles. The process requires many specialized activities and inevitably takes considerable time.

1. Specify or define and then describe the nature of the physical problem, which can often be done relatively quickly using an appropriately annotated sketch. Typically, for an aerodynamic problem, the annotations could include the size and shape of a control volume, the general flow directions, and the specifications of relevant boundary conditions.

2. Mathematically specify any relevant known boundary conditions, such as upstream free-stream conditions. On the surface of a solid body placed in the flow, the flow would not pass through that body, so another boundary condition is that the flow is parallel to the body surface.

3. Decide on the primary form of the needed model, i.e., whether an integral or differential approach is required. For example, if detailed properties are needed at all points, an integral approach is unlikely to be appropriate, and the problem should be approached using a differential model.

4. For aerodynamic problems, decide whether an Eulerian or Lagrangian flow model is required, i.e., whether the aerodynamic behavior at a fixed point or over a volume in space is needed or whether the identical fluid particles need to be tracked as they move through the flow.

5. Make any justifiable assumptions about the problem. The idea here is to take the actual physical problem and derive a simplified but still relevant mathematical version of the physical problem. By drawing on experience or from experiments on similar problems, it may be possible to make reasonable assumptions. For example, for aerodynamic problems, it may be possible to assume that the flow is steady and/or in predominantly two dimensions, which usually results in considerable simplifications of the mathematics. These assumptions are then used to help develop the appropriate governing equations for the model.

6. Use the conservation principles to establish the model’s mathematical form that describes the physical problem. Because there are three physical principles to invoke (i.e., mass, momentum, and energy), most problems will involve three governing equations. However, auxiliary equations (e.g., an equation of state) may also be needed. These equations then need to be solved consistently and concurrently.

7. Conduct the solution process where the relevant equations are solved for the desired physical quantities, e.g., for flow problems, velocities, pressures, etc., may be needed. In some cases, the equations may be solved analytically in closed form, meaning that the resulting solutions are pure mathematics and the creation of final sets of descriptive equations. The equations will likely need to be solved numerically, i.e., a computer program must be written with numbers as the outputs.

8. Verify and validate the model to determine the model’s accuracy and correctness, a process often called “V & V,” i.e., use predicted outcomes to determine how good the model is in representing the physical behavior that was the desired outcome. The model’s validity can be established by comparing the results against measurements if such measurements are already available or can be conducted, as shown in the figure below. This step is essential in aerodynamic modeling and is one reason wind tunnels are critical in understanding all types of aerodynamic flows. If experimental data are unavailable, sometimes other theories can be used for validation, but validation is rarely conducted without reference to appropriate measurements. In all cases, experience and good judgment must be used to establish that the predictive credibility of the model has been obtained. This is rarely a test that one person can objectively conduct because confirmational bias, also known as confirmation bias, can play a role. [2]

9. Improve upon the capabilities of the model to broaden its capabilities. As experience is gained in the validation process regarding what the model predicts adequately and what it does not, the limitations and assumptions within the model can be progressively removed, or other enhancements can be made. For example, extending the range of validity of an aerodynamic model may be possible by including unsteady effects or with a representation of turbulence. In a structural model, it may be necessary to include non-linear effects, e.g., from large deflections.

10. Finally, balance the model’s complexities and capabilities against the time and cost of obtaining solutions from the model. In this case, questions will have to be asked about how the model will be used and whether the full fidelity of the model is needed. For example, the need to model compressibility effects in the flow might not be required if the problem is restricted to low Mach numbers. Furthermore, if the model is to be used exclusively for research, then computational time and cost will be less crucial than for use in the industry, where a short turnaround time is always needed.

Generalization of Data & Data Fitting

$\Pi$

For example, a pure theoretical model may be of the form

$\begin{equation*} \Pi_1 = \Pi_2^{~2} \end{equation*}$

which may show good qualitative agreement with measurements but not quantitive agreement. However, a semi-empirical model of the form

$\begin{equation*} \Pi_1 = A + B \, \Pi_2^{~2} \end{equation*}$

It is rarely possible to cover the entire domain of the parameter space in one single experiment. In this regard, not all test facilities are created equally. One experiment may cover one limited range of the domain, and another experiment may cover another range, perhaps with some overlap, as suggested in the figure below. Each experiment may be inadequate in establishing any useful, robust, causal correlation. However, if the data is taken collectively and used intelligently in the process of generalization, then a semi-empirical model can show a more robust and positive correlation.

Repetition of experiments is always a good practice, which can also benefit from the advancements in measurement technologies. Indeed, the scientific method requires that any one experiment can only be fully trusted once it has been independently repeated and the results confirmed. Unfortunately, repeating experiments is often difficult to justify based on cost, time, or both. Inherent differences in the testing facilities can also be a consideration, i.e., wind tunnel interference effects, measurement limitations, etc. However, the logical rationalization here from the perspective of the scientific method is that the ability to make better measurements will lead to lower errors and inaccuracies in the acquired data. So, a better and stronger causal correlation is likely to be shown. Therefore, more confidence in prediction can then be gained. Nevertheless, for any given dataset, there is always considerable uncertainty in extrapolation beyond the range of the measured data.

Inevitably, new experiments on a given problem are performed in time, and some experiments may be ambitious enough to extend the range of the domain. Such data may confirm existing trends or be disruptive, suggesting a different correlation, as shown in the figure above. On the one hand, such disruptive data leads to an improved correlation and more confident mathematical models that show enhanced predictive confidence at the system level. On the other hand, predictions at the system level may be worse, in which case one can suspect that one modeling error has previously acted to cancel another, i.e., duo mala faciunt unum bonum , and so point to further modeling deficiencies. The stakes in such outcomes are high in that the validity of the entire system model can be thrown into question until the weaknesses in the other parts of the model are identified and corrected. In this regard, complex models, especially those with significant empiricism, can remain perpetually tentative until more and/or confirmatory measurements are obtained.

Ockham’s Razor – The KIS 2 Principle

One issue with complex engineering models for multidisciplinary aerospace applications is that significant empiricism may be needed. Some physical problems are difficult to model without resorting to substantial empiricism, an unavoidable artifact of representing complex physical processes with parsimonious models with practical levels of computational efficiency. In this regard, there is always a need to balance the complexity of the mathematical model against the model’s predictive accuracy while aiming to minimize the variability and maximize the intelligibility of the resulting simulations. For complex mathematical models, history has proved that predictive accuracy increases with increasing modeling complexity only up to a point where the cumulative uncertainties in the components of the model (particularly those with significant empiricism) begin to increase the “noise” in the predictions. Then, beyond a certain level of complexity, the predictive accuracy decreases again, the system exhibiting a classic “Ockham’s Hill,” as shown in the figure below.

In this regard, it is essential to remember the principle of Ockham’s Razor, i.e., given two sets of solutions from methods of equivalent accuracy, one should side with the simpler or parsimonious method, i.e., Frustra fit per plura quod potest fieri per pauciora. This approach is sometimes called the KISS or KIS2 principle, which means “Keep it Short and simple.” Ernst Mach was also an advocate of a similar principle he called the “Principle of Economy,” stating: “Scientists must use the simplest means of arriving at their results and exclude everything not perceived by the senses.” The message is evident in that the goal for engineers is to keep modeling complexity and predictive fidelity in balance, something that cannot just be achieved through careful, systematic validation studies but also requires a degree of common sense.

Approaching Homework Problems

The skills and abilities to solve real engineering problems develop from the exemplar problems encountered in the classroom. To this end, budding engineers must first be good at doing homework problems. Results generated in homework problems may be single numbers, tables, or graphs in any or all combinations that must be adequately presented. Below are some general guidelines to help new students tackle homework problems and build up the skills they need to learn as engineers.

Homework is NOT a quiz!

The idea of homework for engineering students is to begin to learn the process of solving real problems, which becomes a lifelong endeavor for an engineer. Homework problems expand upon what can be done in the classroom, which are usually simpler and straightforward. In doing the homework, students can use all the resources available, including exemplar problems and solutions given in the textbook or ebook, for which many are former exam or quiz questions. Students should use these exemplars to help them understand the process of solving similar problems that may come up on homework and future exams.

If you are a student and need help understanding how to go about a specific homework problem, then ask your instructor before or after class, during office hours, or by email. Refrain from guessing! Also, do not copy down former “solutions” to similar problems and present them as your own – use the exemplar solutions to understand the process of solving the current homework questions, then write out your own solutions. Finally, do not assume that the current question(s) is (are) exactly the same as a previous question you may have seen; submitting the correct answer to the wrong homework problem will not have a good outcome.

A student’s homework score may be a significant fraction of their total grade in an engineering course. This fraction reflects the importance instructors place on learning the methods and techniques of solving engineering problems and developing and maintaining other skills, including presentation formats, time management, MATLAB use, drawing graphs, etc. Remember that students may see exam questions very similar to homework questions, which are often easier! So, if students approach homework problems seriously and put in the required time to understand the processes of solving them, they can expect to do well on the exams. History speaks for itself in this regard.

Pointers for Doing Good Homework

Preparations.

Start to review the homework problem set as soon as the professor or course instructor sets it. Remember: Start working on the problem set early! Time management is essential.
Don’t guess! Homework is not a quiz. It is a learning exercise, so don’t consider homework as a quiz or exam.
If you need help with the homework questions, attend the professor’s, instructor’s, or TA’s office hours. Come prepared and ask specific questions; other students may be waiting to ask their questions.
Don’t post your homework problems online, hoping that someone in cyberspace knows better than the person who wrote the question. Ask the professor or whoever wrote the problem(s) for advice.
Check whether a similar problem is solved in the textbook, the ebook, or the notes. Review old problem sets and their solutions – professors usually make them freely available to students.
Meet with other students, your study group, the TA, the grader, or anyone else and try to understand the problem and potential solution method. An effective learning method is for one student in a study group, for example, to try to teach the solution method to the others.

Starting on the Problems

State the problem statement briefly and concisely based on the information given. It is often helpful to restate the information in the question, clearly and straightforwardly laying out what is known and what is not.
If appropriate, draw a sketch or schematic of the problem/approach. In most cases, an annotated sketch of the physical problem will help you decide the nature of the mathematical model that needs to be adopted, e.g., control volume approach or otherwise.
Write down the appropriate mathematical equations necessary to solve the problem. These equations should not be expected to be given in the homework problem, at least not in all cases. Choosing (or deriving) a set of simpler equations from a broader, more general set of governing equations may be necessary.
List and develop any simplifying assumptions appropriate to the problem. Sometimes, the assumptions will be specified; other times, they may be left as part of the problem. For some more challenging problems, it may not be apparent initially what assumptions are needed. Several attempts may be necessary until the correct assumptions can be confirmed and verified.

Working the Problems

Be sure to work carefully and systematically through each of the problems. It is better to work slowly but get the correct answer than to work faster and make silly mistakes.
Complete the analysis in algebraic form (i.e., equations with symbols and variables) before substituting the specific numerical values. Sometimes, the answer will be presented as an equation rather than a numerical value.
Substitute known numerical values (using a consistent set of engineering units ) to obtain a numerical answer or answers. If the problem is given in SI units, it is best to work it entirely in SI units; conversely, if it is given in USC units, then work it entirely in USC units. For example, switching back and forth between USC and SI is inadvisable because this approach is often a source of mistakes and numerical errors.
All numerical answers must have appropriate units unless they are in non-dimensional form. Always double-check the engineering units of the solution (s). Units should be used consistently throughout, ideally in base units.
Check that the number of significant digits in the answer(s) is/are consistent with the given data. For example, if you are given information to 3 significant digits, it would not be appropriate to calculate your final results to 5 significant digits. However, it is good practice to round off numerical values at the end of the problem.
Review the answer(s) for correctness. In some cases, it will be evident if the result is wrong; it may be challenging in other cases. In many cases, the question is: Does that result seem physically correct? Check with someone else if in doubt, such as a course instructor.
Draw a box around the final answer to clarify that this is your definitive answer. The answer will often be an equation (formula) or number, but it could be a table or a graph. You do not need to draw a box around tables or graphs.
Import the results into the appropriate software if a graph needs to be drawn. Never draw graphs freehand!! Use MATLAB, Excel, or your favorite graphing program. Kaleidagraph was used for many of the plots in this ebook. As appropriate, all graphs need to have proper legends, labels, and other annotations.
Your submitted homework must be neat and easy to follow. You will also need to follow the submission rules. For example, you may have to use squared engineering paper, put your name and student identifying number on each page, staple together your pages, etc. If not, you will likely get less credit regardless of the correct solution. In industry, great emphasis is placed on the clarity and presentation of reports and papers; for the same reasons, clarity in homework is a place to start.
Write down clearly and unambiguously the names of the student(s) you worked with on the homework, if any. Working with others is acceptable , but you should submit your OWN answers to the questions. For example, you may state, “I cross-checked my final answers with John and Kevin, and my answers agreed with theirs.”

All those Equations!

In engineering (or any science field), students do not have to memorize hundreds of equations or “formulas.” The questions and concerns from new students of the field inevitably soon start to flow, such as: ” What equation do I use?” or “Where do I find the equation?” or “What is this symbol in this equation?” or “Where do I get the solution to this integral?” Such questions are natural, and guidance from experienced engineers and professors is essential. As taught in many educational contexts, the “plug and chug” paradigm of plugging a numerical value into some equation and chugging out an answer is a surefire recipe for disaster in actual engineering problem-solving. Bona fide engineering students need to learn to do much better.

The number of equations in science and engineering could be as many as the number of stars in the galaxy! No exaggeration. As the figure below suggests, most successful engineers and professors remember the details of the biggest stars and the most general form of the equations or the “governing” equations. They also understand the concepts and have done enough problems to know how to reduce and simplify the equations under certain assumptions and conditions to produce subsets of equations with specific applicability to the problem at hand. The equations they can’t remember can usually be derived! The governing equations can then be adapted and applied successfully to various situations by grasping the underlying concepts without memorizing dozens of problem-specific equations. Science and engineering is not a memory game.

The risk in memorizing without understanding is that the wrong equation (or equations) is (are) applied, so the answer obtained is inevitably wrong too, which is always a disastrous outcome in engineering problem-solving. Therefore, rather than rote memorization or hunting down a specific formula that may or may not be applicable, understanding the fundamental principles allows engineers to derive and apply the relevant equations to particular problems. While some foundational equations may be essential at one’s fingertips, the focus should still be on understanding the underlying physical principles and fundamental concepts expressed by the equations, appreciating the meaning of all the terms in the equations, and rationalizing the best solution strategies for these equations.

Moreover, with access to the Internet, lots of information and resources are readily available for students to find the more general form of the equation(s) that might be needed. Unfortunately, there is a lot of opinionated absurdity on the Internet, especially regarding engineering topics, so using authoritative resources is critical. Utilizing selective online resources, as well as peer-reviewed publications, textbooks, and sanctioned computational tools, can help new students and practicing engineers retrieve or verify specific equations and/or solution methods when needed. Labora sapienter, non strenue.

What is Brainstorming?

Brainstorming is an informal but highly effective approach to engineering problem-solving that works for homework problems. Usually conducted with a small group of engineers or students and a single moderator, the process encourages all participants to think laterally and develop ideas that might initially seem unusual or even sound slightly crazy!

A group of five to seven people is usually the most influential, with a mix of experienced and less experienced engineers. The moderator should be someone other than the chief or lead engineer or an engineer in management, and the group itself should select the moderator. The people in the group should come from several technical disciplines to foster and develop the most effective and productive brainstorming environment. Excellent ideas may ultimately flow from engineers who see an avenue of opportunity in a discipline different from their own, i.e., when they start looking at the problem with a fresh mind. Some of these ideas may be developed into a rational basis for engineering problem-solving, often following a new or innovative path that nobody else had considered previously. The basic idea is to get all participants to think “out of the box,” be creative, and divert their attention away from using their “conventional wisdom” to solve problems.

Brainstorming is usually very effective for solving complex engineering problems requiring multi-disciplinary engineering. The best ideas in a brainstorming session will often flow from the less experienced engineers, who are not so encumbered by conventional wisdom. Brainstorming is best conducted in an informal, relaxed environment away from the typical day-to-day work environment, often at a retreat location. Brainstorming can also be fun, an excellent environment for team building, and a way of getting to know other engineers outside one’s primary technical discipline or organization. It is not unusual for companies to work together on solving complex engineering problems in the aerospace field, and brainstorming sessions can foster more substantial inter-company dialog where everyone works more effectively together.

For brainstorming to be effective, all group members must be active participants, and personal criticism is inappropriate. Often, some quirky idea from one group member makes no sense at first, but after discussion, there can be an “aha!” or “I never thought of that!” moment, and the idea can be built upon by the group after that. Alternatively, the idea may lead to some other revelation and a different path to solving the problem that nobody thought about before.

In preparing for the brainstorming session, a location with no distractions must be found, all phones and computers should be turned off, the doors locked, and a whiteboard should be available to write down the ideas. Everyone should have an opportunity to speak when they want to. The moderator must refrain from allowing any one member of the group to dominate the brainstorming session. At the end of the session, the group decides on the best ideas and then moves forward, as needed, to follow up and pursue them. The history of engineering suggests that many of the best and most innovative ideas can come from brainstorming sessions.

Summary & Closure

Developing mathematical models that can be used to study and solve various problems is an integral part of engineering. However, the selection process must be conducted carefully and systematically to choose or develop the relevant governing equations that apply to the specific problem (or problems) of interest. In some cases, it may be possible to down-select the appropriate governing equations for a particular problem from more general forms of governing equations that are intended to apply to a broader range of conditions. In other cases, the governing equations may need to be developed from first principles.

Once a mathematical model or set of models has/have been developed, it is necessary to solve the equations to make predictions about the system being modeled. The solution process can involve analytical methods, such as closed-form solutions, or numerical methods, such as finite element, finite difference, or boundary element methods. The choice of solution method will depend on the nature of the problem, the available resources, and the desired accuracy of the solution. In all cases, the justification of assumptions and/or approximations is needed. In this regard, any reason may require reliance on outcomes from experiments, i.e., for verification and validation of the modeling. A combination of solution methods may often arrive at an acceptable solution. Also, post-processing and visualization techniques may help interpret the results and behavior of the modeled system.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

The “KISS” or “KIS 2 principle refers to the acronym for “Keep It Short & Simple.” Discuss the meaning of this principle as it might apply to engineering modeling.
What is a parametric study, and why might we conduct one in engineering design?
How might assumptions impact the accuracy of models? Can overly simplifying a problem lead to inaccurate results
What might be the trade-offs between making assumptions to simplify a model and retaining complexity for better accuracy?
Why is it essential to validate and verify models? What are some standard methods for validation, and how do they contribute to model credibility?
How might advances in computational power and simulation techniques influence how we set up and solve flow models in the future?

Additional Online Resources

An excellent video on the use of mathematical and computer models in engineering.
Video on the use of models and simulation in engineering.
View a video on Eulerian and Lagrangian flow models.
Navigate here to watch a video from the National Science Foundation on types of flow models.
Review the KIS 2 Principle here and a video here .
The hypothetico-deductive method is a fundamental approach used in scientific inquiry across various disciplines, guiding the systematic investigation of natural phenomena and developing scientific theories. It emphasizes the importance of empirical evidence, logical reasoning, and rigorous testing in the process of scientific discovery. ↵
Confirmational bias is a cognitive bias where individuals tend to favor information that confirms their existing beliefs or hypotheses while disregarding or downplaying contradictory evidence. In other words, people tend to seek information that supports what they already believe and ignore information that contradicts it. This bias can lead to flawed decision-making and judgment because it can prevent individuals from critically evaluating evidence and considering alternative viewpoints. ↵

Introduction to Aerospace Flight Vehicles Copyright © 2022, 2023, 2024 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License , except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n15

Department of Engineering Education >
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Research Projects

chalkboard with equations and bookshelf.

Our faculty members conduct research in the areas of Learning Experiences, Pedagogical Innovations, and Systems and Institutions—with projects focusing on topics such as student learning, curricular design, student success and assessment techniques.

Featured Research Projects

Interested in working with our faculty members on one of these research projects? Contact us with your interest .

Developing Homework Problems to Increase Conceptual Knowledge Development and Sense-making

Principal Investigator: Jessica Swenson

Abstract: Engineering students spend a significant amount solving homework problems for their technical, core courses. Yet, we know little about what students are doing as they solve these homework problems. Dr. Swenson’s previous work examined student group discourse as they solved assigned homework problems and found students conversations mostly focused on getting problems done instead of discussing concepts and their application. This project will focus on developing homework problems that emphasize making sense of concepts, especially through writing and discussion. Work on this project will include collecting data on current homework problems, developing prompts, and piloting problems with small groups of students.

Impact of Pre-College Computing Education

Principal Investigator: Adrienne Decker

Abstract: There are many initiatives in place to introduce computing (and programming) to audiences before reaching university-level education. This project investigates the landscape of pre-college computing education and its impact on those participating. The project is centered around a resource center which contains curated information about pre-college computing education as published in the literature, a set of validated evaluation instruments for computing education, and resources to introduce newcomers to research in this space. The continued development of this resource center is one aspect of the project, but another is the analysis of the data contained within and the collection of new data about the impact of these activities on participants.

Investigating the Role of Problem Typology in Helping Engineering Undergrads Effectively Communicate Their Experience

Principal Investigator: Andrew Olewnik

Abstract: An important and recognized challenge for undergraduate programs is to provide engineering students with experiences that provide insight on what it means to be an engineer in practice. For such experiences to be truly meaningful to professional formation, students must also be capable of internalizing and effectively communicating insights from these experiences later. In this research to “internalize and communicate experience” refers to a student’s metacognition of their professional competencies and the level to which they can effectively communicate those competencies. This ongoing project explicates a problem typology and reflection framework as context for student problem/project-based learning experiences. Through mixed methods research that includes group problem solving discussions, written reflection, and mock interviews, we are investigating the role of problem typology in helping students to: (i) recognize and orient themselves to different types of engineering problems; (ii) deconstruct and re-synthesize technical experiences in terms of specific professional competencies; and (iii) improve communication of professionally relevant experiences to external parties - i.e. employers.

Engineering education project database map.

Open-ended Modeling Problems for Engineering Science Courses

Principal Investigator: Jessica Swenson

Abstract: The Open-ended Modeling Problems project is working to create new homework problems for technical, core engineering courses that are open-ended (no one correct answer) in nature. These problems ask students to use course content to create a mathematical model of a given real-world scenario. The study examines how students approach these problems as well as how it is developing their engineering judgment. Work on this project includes analyzing previously collected interviews, in-class discussions, and students’ homework, as well as helping the growing research team to develop new open-ended modeling problems.

airplane with mathematical equations pointing to wheels.

Problem Typology as a Foundation toward an Engineering Education Problem Database

Abstract: Engineers are known for defining themselves as problem solvers, and solving open, complex problems is recognized as exemplary practice. However, there is no agreement on how an expert behaves in practice, nor is there agreement on specific problems, protocols, or rubrics to assess student learning as they work toward becoming expert problem solvers. Using engineering problem typology and problem solving characteristics described in the literature, this research seeks to develop a standard for categorizing problems along dimensions like structured-ness, complexity, representation, and domain knowledge. This research requires collaboration and investigation across academics disciplines and with experts in the field in order to contribute to our understanding of specific differences and commonality between disciplines at the resolution of the individual stages of the different engineering problem types. Such knowledge could help to inform the pedagogical approaches, assignments, and assessment methods in individual courses, and serve as a foundation for a standardized, community-developed database of engineering problems.

Use of Homework in Problem-Solving Courses

Principal Investigator: Carl Lund

Abstract: Multiple iterations of practice and feedback are effective for the development of problem-solving ability. Often in engineering courses each homework assignment is graded and used to assess achievement of course learning outcomes. Feedback to students occurs too late and may be limited to providing a “correct” solution to the homework assignment and, perhaps, a few terse comments written on the submission. Alternative approaches to assigning homework that afford opportunities to fail, receive feedback and learn from mistakes prior to assessment of learning are being studied in this project. These approaches include scaffolded in-class practice, grading initial assignments only on effort and using homework wrappers to better target feedback, combined with explicit instruction of problem-type identification and general solution strategy.

Using Subgoal Labeling to Teach Introductory Programming

Principal Investigator: Adrienne Decker

Abstract: This project involves the introduction of subgoal labels into the introductory programming class as a pedagogic technique for enhancing student experience and learning. The research team has developed worked examples of problems that incorporate subgoal labels, which are explanations that describe the function of steps in the problem solution to the learner and highlight the problem solving process. Using subgoal labels within worked examples, which has been shown effective in other STEM fields, is intended to break down problem solving procedures into pieces that are small enough for novices to grasp. The developed subgoals are being piloted at various academic institutions in the US in the upcoming academic year. The team is now focused on analysis of the data from the pilot sites and working on issues of deployment and dissemination. In addition, subgoal labels for additional courses beyond the first course are being explored.

Technical Communication Research

SEAS at UB considers communication skills central to the success of their students, and the Department of Engineering Education houses some of the best teacher-scholars in technical and scientific communication. Faculty cover a range of disciplinary expertise, and students have the opportunity to participate in undergraduate and graduate research project focused on communication in engineering contexts.

Technical Communication research projects in DEE include:

Cross-cultural Usability (Keshab Acharya)
Communicating with the Public in Transportation Projects (Kristen R. Moore)
Introducing Standards into the STEM Communication Curriculum (Lauren Kuryloski and Kristen R. Moore)

Problem Solving and Mathematical Modeling

First Online: 11 January 2023

Cite this chapter

Parikshit Narendra Mahalle 7 ,
Nancy Ambritta P. 8 ,
Sachin R. Sakhare 9 &
Atul P. Kulkarni 10

Part of the book series: Studies in Autonomic, Data-driven and Industrial Computing ((SADIC))

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A problem is a puzzle or task that requires a logical thought process or fundamental mathematical steps to solve it. The puzzle generally represents the set of questions on the underlined use case which also consist of complete description of the use case along with the set of constraints about the use case. Logic is very importantly used to solve the puzzle or problem. Logic is defined as a method of human thought that involves thinking in a linear, step-by-step manner about how a problem can be solved. Logic is a subjective matter and varies from person to person. Logic is directly linked to the natural intelligence of human being and is a language of reasoning. Logic also represents the set of rules we use when we do reasoning. It is observed that majority of the employment of fresh engineering graduates across the globe is in software and information technology sector.

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Joshi SA, Mahalle PN (2016) Design and analysis of algorithms: a problem solving approach. Cambridge University Press, India, p 209

TOCICO Board of Directors—Theory of constraints international certification organization. www.tocico.org . Retrieved 17 Mar 2018

Dettmer HW (2007) The logical thinking process: a systems approach to complex problem solving. ISBN: 978-0-87389-723-5

Pólya G (1945) How to solve it. Princeton University Press. ISBN: 0-691-08097-6

Dormy RG (2005) How to solve it by computers. Prentice Hall

Levitin A (2005) Introduction to design and analysis of algorithms. Pearson Education Press

Frejd P (2014) Modes of mathematical modelling: an analysis of how modelling is used and interpreted in and out of school settings. Linköping University

Blum W (2015) Quality teaching of mathematical modelling: what do we know, what can we do? In: Cho S (eds) The proceedings of the 12th international congress on mathematical education. Springer, Cham. https://doi.org/10.1007/978-3-319-12688-3_9

Ball DL, Hill HC, Bass H (2005) Knowing mathematics for teaching. Am Educ 29(3):14–46

Biccard P, Wessels DCJ (2011) Documenting the development of modelling competencies of grade 7 students. In: Kaiser G et al (eds) Trends in teaching and learning of mathematical modelling (ICTMA 14). Springer, Dordrecht, pp 375–383

Chapter Google Scholar

Kantner J (2008) The only absolute truth in mathematics is the myth of mathematics as universal. Perspect: N Y J Adult Learn 6(2). Retrieved from http://www.eric.ed.gov/PDFS/ED501486.pdf

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Mahalle, P.N., Ambritta P., N., Sakhare, S.R., Kulkarni, A.P. (2023). Problem Solving and Mathematical Modeling. In: Foundations of Mathematical Modelling for Engineering Problem Solving. Studies in Autonomic, Data-driven and Industrial Computing. Springer, Singapore. https://doi.org/10.1007/978-981-19-8828-8_2

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Title: pecc: problem extraction and coding challenges.

Abstract: Recent advancements in large language models (LLMs) have showcased their exceptional abilities across various tasks, such as code generation, problem-solving and reasoning. Existing benchmarks evaluate tasks in isolation, yet the extent to which LLMs can understand prose-style tasks, identify the underlying problems, and then generate appropriate code solutions is still unexplored. Addressing this gap, we introduce PECC, a novel benchmark derived from Advent Of Code (AoC) challenges and Project Euler, including 2396 problems. Unlike conventional benchmarks, PECC requires LLMs to interpret narrative-embedded problems, extract requirements, and generate executable code. A key feature of our dataset is the complexity added by natural language prompting in chat-based evaluations, mirroring real-world instruction ambiguities. Results show varying model performance between narrative and neutral problems, with specific challenges in the Euler math-based subset with GPT-3.5-Turbo passing 50% of the AoC challenges and only 8% on the Euler problems. By probing the limits of LLMs' capabilities, our benchmark provides a framework to monitor and assess the subsequent progress of LLMs as a universal problem solver.

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How to successfully utilize MTSS problem-solving models in education

March 18, 2022

Yvette Arañas, NCSP

Dr. Rachel Brown, NCSP

Utilizing MTSS problem-solving models in education is the best way to help students who are struggling in school. So, what does the problem-solving process look like?

In this blog, we’ll explain how to approach MTSS problem-solving models in education with Renaissance’s five-step problem-solving method.

How to approach MTSS problem-solving models in education

When a school implements an MTSS model , how can educators figure out what type of support each student needs and whether a student should receive intensive, one-on-one interventions? One approach that schools can take is to have a problem-solving team.

Such teams can help teachers plan and implement tiered interventions and collect data to identify which students need more intensive academic or behavioral support. Usually, a referral is made to the problem-solving team when a student is not improving despite receiving Tier 1 (e.g., core) plus Tier 2 (e.g., small group) intervention.

What does a problem-solving team look like?

The members of a problem-solving team (PST) act as consultants to teachers and other staff who have worked with a student. Because many of the referrals could reflect various academic and behavioral problems, it is important to make sure that the team is made up of people from different educational positions who can offer different perspectives about students’ needs.

Ideally, a PST should include at least one classroom teacher, a special educator, a school psychologist, a counselor or social worker, and the building principal. All PST members should be encouraged to think about students’ school difficulties in relation to the expectations of students in the same grade level. The primary function of the PST is to reduce any differences between those expectations and the student’s current school performance.

Responsibilities of the PST

The PST is the engine that drives the MTSS system. By reviewing school-wide data, the PST can proactively address system needs and support individual student growth.

The team members should meet regularly throughout each month with a structured agenda that varies to:

Review universal screening data
Provide expertise related to MTSS professional development
Review school-wide data and make data-based decisions
Collaborate with general education teachers to support grade levels/departments serving students during interventions
Provide expertise regarding school site enrichment/intervention schedule, course offerings, and curriculum
Communicate MTSS-relevant issues to the site administrator
Hold problem-solving meetings with parents for individual students
Collaborate and consult with teachers, counselors, administrators, and parents about the MTSS problem-solving model
Review data and refer students for comprehensive special education evaluations when warranted

Communication and collaboration is critical to the functioning of an effective PST.

What is an MTSS problem-solving model?

The MTSS problem-solving model is a data-driven decision-making process that helps educators utilize and analyze interventions based on students’ needs on a continual basis.

Traditionally, the MTSS problem-solving model only involves four steps:

Identifying the student’s strengths and needs, based on data.
Analyzing data and formulating appropriate interventions.
Implementing these interventions.
Reflecting on and evaluating intervention outcomes.

Let’s consider how this compares to Renaissance’s five-step MTSS problem-solving model.

Renaissance’s five-step MTSS problem-solving model

While similar to the traditional four-step MTSS problem-solving model, the Renaissance problem-solving method adds an extra step to fully flesh out the process. Renaissance endorses a five-step problem-solving method that includes:

Problem Identification
Problem Analysis
Plan Development
Plan Implementation
Plan Evaluation

When used at any time of the school year, MTSS problem-solving models in education help school teams engage in data-based decision-making for the benefit of all students.

Let’s consider each of the components in detail.

#1: Problem identification

Every educator knows that a student can develop school-related problems at any time of the school year. Problem identification is the point in time when a possible problem first shows up on the “radar” among school staff.

For example, in the spring, especially before the final screening period, problems that may arise include issues like:

Students who have not (yet) responded to intervention
Students who were doing well earlier in the year but are now struggling with more challenging material
Efforts to have 80% or more of students meet the winter learning benchmarks were not achieved

In each of these cases, the next step is to analyze the problem based on available evidence.

#2: Problem analysis

Problem analysis involves using collected data to identify the size and effects of the problem. Some problems might be small enough that they do not justify additional resources to address.

For example, if a student’s progress data indicated somewhat lower scores after a school break, but quickly returned to higher levels, giving attention to this probably does not make sense because the student’s performance improved once school resumed.

A bigger problem could be if many students in a class or grade did not meet a benchmark screening goal. It is expected that all students will make growth throughout the school year.

To recognize this growth, benchmark goals are adjusted upwards for each screening period during the school year. If there are students who met the benchmark in the fall but not the winter, more information to analyze the source of the problem is needed.

Examples include:

What percentages of students did and did not meet the goal?
Were they all from the same class or different classes?
Did they have similar scores in the fall?
How close to the goal are they now?
Exactly what instruction has been provided to these students?

To address the above questions and complete a thorough analysis, additional data about student performance and instructional practices might be needed. The most recent (e.g., winter) screening scores answer the first four questions. An interview with teachers and/or classroom observations could answer the final question.

The goal is to develop a hypothesis about exactly why an unexpectedly large number of students did not meet the winter benchmark.

Implementing MTSS strategies

Discover tools from Renaissance that help you to identify and meet every learner’s needs.

#3: Plan development

With a hypothesis in hand, the school team then turns to consider possible plans that can address the problem. The planning process needs to cover what steps will be taken to improve the scores of the underachieving students.

One approach could be to add those students to existing intervention groups so that they can participate in additional instruction . This might work if the number of students is small, but if many students need intervention, there might not be enough groups or interventionists to teach them.

Another approach would be to revise the Tier 1 core instruction to include the specific skills that these students lack. This approach has the benefit of meeting the needs of more students at one time, but it will work only if the students have relatively similar learning needs.

The team might need to explore both of these options and then compare the costs and benefits of each to make a decision.

In addition to developing a short-term solution to the students’ learning gaps, the team should think about and work on plans to develop a way to prevent the same thing from happening next year. Collecting and using continuous data to set longer-term goals is an important part of system-level data-based decision-making.

If the team checks and reviews the effects of the selected plan to support the struggling students throughout the rest of the school year, the information gained can help in a decision about whether additional new plans are needed.

Specifically, in the course of observing the effects of either small-group interventions or whole-class instructional changes, the team will likely have more information about whether the original Tier 1 core instruction that appeared to work in the fall needs to be changed in the future.

Such changes could include additional training for teachers, a new pacing guide that narrows the focus of what to teach, or an entirely new set of materials and methods.

Using group-level data from the current school year is an essential and effective way to plan for the future needs of all students.

#4: Plan implementation

Once a specific plan for addressing the student’s needs is developed, the next step is to implement it. For any plan, additional resources will likely be needed.

For example, if the students are added to existing intervention groups, who will gather and prepare the necessary materials? If the Tier 1 core instruction is changed, how will teachers learn about the changes and become ready to teach the revised lessons?

To support those charged with plan implementation, it’s very helpful for a member of the school’s problem-solving team to conduct regular “check-ins” with those implementing the plan. These checks can be weekly and will help the staff know that they are supported in their efforts to meet individual students’ needs.

Another component of implementation is to verify intervention or teaching accuracy. This is also called teaching integrity, or fidelity. It is important because it provides data about whether the planned change was done correctly. Checking on teaching integrity can involve interviewing the teachers or observing lessons.

The method used should match the…

Location; and

…of the instruction.

Having teaching integrity is important because unless the data collected as part of the instructional change can be trusted, there is no point in evaluating the plan.

#5: Plan evaluation

The final step of the problem-solving model is to assess the data collected during and after the changes to see whether they worked.

In the case of instructional changes made between the winter and spring benchmark periods, one way to evaluate the plan would be to see if the students’ spring screening scores reflect a significant improvement over the winter scores.

The downside of relying entirely on the spring screening scores is that they might not be collected for weeks or months. Instead, it could be better to gather additional data on student performance before the spring screening.

If the team opted to have the students join existing intervention groups, the newly added students should complete regular progress measures alongside all the other students in the groups.

Given the time of the year and the urgency of the learning needs, weekly progress monitoring would be recommended. If the team opted to change the Tier 1 core instruction, alternate assessments might be better, depending on whether there are brief and easy-to-administer options that all students could complete.

Renaissance offers two solutions—FastBridge and Star Assessments— that can be used for regular progress monitoring and whole-class interim assessments.

FastBridge assessments for reading that would work for both small and large groups include AUTOreading and COMPefficiency. For math , FastBridge offers CBMmath-Automaticity and CBMmath-Concepts and Applications (CAP).

These are all computer-administered and scored assessments that provide immediate feedback on student performance. For younger students, selected measures from FastBridge earlyReading and earlyMath can also be used for progress monitoring as they are used at least monthly.

Similarly, the Star suite includes both computer-adaptive assessments (CATs) and curriculum-based measures (CBMs) for reading and math. Star CATs and CBMs support progress monitoring and whole-class interim assessment in both English and Spanish, from preschool through grade 12.

Utilizing MTSS problem-solving models with Renaissance

Both FastBridge and Star Assessments are aligned with a problem-solving approach to assisting students.

The five problem-solving steps can be used continuously throughout the school year to identify, define, and address individual and group learning needs. Educators do not need to wait until the new school year before addressing such learning needs.

Instead, they can continue to use the problem-solving steps with new issues as they arise. Using the problem-solving approach throughout each school year has benefits for both individual students and groups:

At the individual student level, the benefit is improvement in skills and a more satisfactory school experience.
At the group (e.g., grade or school) level, the benefit is planning for improved system-level practices and will make both students’ and teachers’ school experiences better in the future.

To learn more about FastBridge or Star Assessments and how they support the MTSS problem-solving model, connect with an expert today.

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Solon Schools' Future Problem Solving teams send 24 students to International Bowl

Updated: Apr. 30, 2024, 12:47 p.m. |
Published: Apr. 30, 2024, 12:14 p.m.

The Solon High School Science Olympiad team won the state championship April 26-27 in Columbus to qualify for the national tournament. Team members are pictured above. (Photo Courtesy of Solon City Schools)

Ed Wittenberg, special to cleveland.com

SOLON, Ohio -- Future Problem Solving teams from Solon High School, Solon Middle School and Orchard Middle School excelled at the State Bowl hosted by Solon recently.

On Monday (April 29), Superintendent Fred Bolden told the Solon Board of Education that 24 Solon Schools students qualified for the International Future Problem Solving Bowl June 5-9 at Indiana University in Bloomington.

• Abnar Fatima, Tina Wang, Maahi Bandi, Rebecca Jacob, Nithya Natesan, Nandita Srikumar, Hiba Patel, Yuvha Karthikeyan, Amritha Kishan, Sanvi Singh, Evan Dan, Frederick Song, Grant Lee and Yuan Kong of Solon High School

• Evan Bai, Mannat Dhooria, Vedanshi Bhatt, Rajanya Bishi and Sofia Patro of Solon Middle School

• Ammar Shah, Mihika Sthul, Japman Kaur, Svanik Kolli and Michael Zhang of Orchard Middle School.

“It’s a very impressive group of kids that did a great job,” Bolden said.

“Kudos to our coaches. They did a fabulous job preparing the kids, and we’re excited to see what they do at nationals.”

Sabrina Tirpak coaches the high school team. Gayle Hauptman coaches the Solon Middle School team and Tanya Perez coaches the Orchard team.

“We are so proud of all of our Solon competitors,” the coaches said in a news release. “The hard work and commitment show in the success our teams demonstrated throughout the competition.”

Overall, Solon students won the following awards at the state meet:

In Community Problem Solving, the Solon High School team of Maahi Bandi, Rebecca Jacob, Nithya Natesan, Nandita Srikumar, Hiba Patel, Yuvha Karthikeyan and Amritha Kishan captured first place.

In Global Issues Problem Solving, the Solon High School Senior Division (grades 10-12) team of Evan Dan, Frederick Song, Grant Lee and Yuan Kong placed second, and the Senior Division team of Maahi Bandi, Rebecca Jacob, Nithya Natesan and Nandita Srikumar placed third.

The Solon High School Middle Division (grade nine) team of Eve Abounader, Sharvi Deshpande, Iris Li and Manasvi Gurajula also placed third.

Solon Middle School’s Global Issues Problem Solving team of Mannat Dhooria, Vedanshi Bhatt, Rajanya Bishi and Sofia Patro earned first place.

Orchard’s Global Issues teams of Mihika Sthul, Japman Kaur, Svanik Kolli and Michael Zhang placed second; Evelyn Kim, Serena Ghazarian, Mahayra Rai and Josephine Josy placed fourth; and Aria Anders, Isabella Capizzani, Olivia Dong and Yusra Fathima placed fifth.

In Presentation of an Action Plan, the Solon High School Senior Division team of Ian Stewart, Aaron Wei and Samuel Kim captured first place, and the Senior Division team of Max Goldberg, Drew Yagour and Zachary Han placed second.

The high school’s Middle Division team of Kaavya Guila, Sophia Shong, Bryna Clayman-Smith and Falak Dahiya took first place, and its Middle Division team of Eve Abounader, Sharvi Deshpande, Iris Li and Manasvi Gurajula placed second.

Solon Middle School’s Presentation of an Action plan team of Mannat Dhooria, Vedanshi Bhatt, Rajanya Bishi and Sofia Patro placed third.

Orchard’s team in that category of Evelyn Kim, Serena Ghazarian, Mahayra Rai, Josephine Josy and Sophia Chen placed second.

In Global Issues Individual Problem Solving, Solon High School’s Abnar Fatima and Evelina Glusauskas captured first and second place, respectively; Solon Middle School’s Evan Bai and James Zhang took first and third, respectively; and Orchard’s Ammar Shah, Angela Xiao and Joyce Lou placed first, second and third, respectively.

In Scenario Writing, Solon High School’s Mandi Lu placed second in the Senior Division, and Tina Wang and Iris Li placed first and third, respectively, in the Middle Division.

The Solon Middle School Science Olympiad team also qualified for the national tournament by capturing the state championship. Team members are pictured above. (Photo Courtesy of Solon City Schools)

Science Olympiad state champs

In other action, the board approved an overnight trip for the Solon High School Science Olympiad team to travel to Michigan State University in East Lansing, Mich., May 23-26 for the Science Olympiad National Tournament.

The Solon High School and Solon Middle School Science Olympiad teams both won state championships April 26-27 at The Ohio State University in Columbus.

Bolden said the board will likely vote on an overnight trip to the national tournament for the Solon Middle School team at its next meeting at 6 p.m. May 13.

Bolden said the Solon High School Science Olympiad team tied the state record for the best possible score -- set by Solon 10 years ago.

“So this is one of the best teams we’ve ever had going into nationals,” he said. “This year, we had a team place in all 23 events, and that’s the first time that’s ever happened.

“It just shows how impressive that this group is and what an amazing job they did.”

In its competition, the Solon Middle School team posted a score of 76, and the second-place team was at 150. Lower scores are better, Bolden said.

“So they absolutely dominated the competition, both middle school and high school,” he said.

“We’re very excited to see what they’re going to do at the national tournament this year. It’s a lot of hard work by the coaches and the kids.”

National Honor Society

In addition, 58 Solon High School juniors were inducted into the National Honor Society April 23.

Students were selected based on their accomplishments in the areas of scholarship, leadership, service and character, Bolden said.

The new Solon High School NHS members are Vennela Siri Appari, Mariana Barrera Gonzalez, Daniela Benitez De Jesus, Jessica Beres, Shelby Berry, Keira Bomeli, Henry Bradt, Alexis Choi, Maggie Coggin, Maggie Cruickshank, Victoria Dai, Riddhima Deb, Rishith Doddi, Kailani Farivar, Abnar Fatima, Alyssa Feldman, Ashlyn Fitzgerald, Shayna Friedman, Anvita Gandhi, Alyssa Gerome, Leah Girzhel, Katelyn Henline, Josephine Howard, Wyatt Johnson, Luke Kim, Chanyoung Lee, Josie Liao, Selena Liao, George Libecco, Isabella Liu, Olivia Liu, Mandi Lu, Lillian Maslona, Madelyn Maslona, Benjamin Mitchell, Tyler Morrison, Olivia Nath, Sneha Nayak, Maya Nayar, Hansa Omur, Qi Pan, Ahalya Pandian, Satyarpita Parameswaran, Mia Patriarco, Rachel Pawlicki, Shawna Polster, Isabel Pulido Gonzalez, Jiya Rai, Aizah Shahbaz, Julia Shao, Keshav Sivaram, Avneet Sohi, Zachary Stein, Kayla Streem, Sushmita Sudhan Supriya, Nina Van Zandweghe, Zaynah Wahab and Julia Wang.

Solon High School’s National Honor Society chapter was founded 73 years ago. Since then, more than 2,500 SHS students have met the rigorous criteria for membership, Bolden said.

Mohammad Asjad

Asjad is an intern consultant at Marktechpost. He is persuing B.Tech in mechanical engineering at the Indian Institute of Technology, Kharagpur. Asjad is a Machine learning and deep learning enthusiast who is always researching the applications of machine learning in healthcare.

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Intervention based on science of reading, math boosts comprehension, word problem-solving

Students working on math problems at a chalkboard.

Mon, 04/29/2024

Mike Krings

LAWRENCE — New research from the University of Kansas has found an intervention based on the science of reading and math effectively helped English learners boost their comprehension, visualize and synthesize information, and make connections that significantly improved their math performance.

The intervention, performed for 30 minutes twice a week for 10 weeks with 66 third-grade English language learners who displayed math learning difficulties, improved students’ performance when compared to students who received general instruction. That indicates emphasizing cognitive concepts involved in the science of reading and math are key to helping students improve, according to researchers.

“Word problem-solving is influenced by both the science of reading and the science of math. Key components include number sense, decoding, language comprehension and working memory. Utilizing direct and explicit teaching methods enhances understanding and enables students to effectively connect these skills to solve math problems. This integrated approach ensures that students are equipped with necessary tools to navigate both the linguistic and numerical demands of word problems,” said Michael Orosco, professor of educational psychology at KU and lead author of the study.

The intervention incorporates comprehension strategy instruction in both reading and math, focusing and decoding, phonological awareness, vocabulary development, inferential thinking, contextualized learning and numeracy.

“It is proving to be one of the most effective evidence-based practices available for this growing population,” Orosco said.

The study, co-written with Deborah Reed of the University of Tennessee, was published in the journal Learning Disabilities Research and Practice.

For the research, trained tutors developed the intervention, developed by Orosco and colleagues based on cognitive and culturally responsive research conducted over a span of 20 years. One example of an intervention session tested in the study included a script in which a tutor examined a word problem that explained a person made a quesadilla for his friend Mario, giving him one-fourth of it, then needed to students to determine how much remained.

The tutor first asked students if they remembered a class session in which they made quesadillas, what shape they were and demonstrated concepts by drawing a circle on the board, dividing it into four equal pieces, having students repeat terms like numerator and denominator, and explaining that when a question asks how much is left, subtraction is required. The students also collaborated with peers to practice using important vocabulary in sentences. The approach both helps students learn and understand mathematical concepts while being culturally responsive.

"Word problems are complex because they require translating words into mathematical equations, and this involves integrating the science of reading and math through language concepts and differentiated instruction," Orosco said. "We have not extensively tested these approaches with this group of children. However, we are establishing an evidence-based framework that aids them in developing background knowledge and connecting it to their cultural contexts."

Orosco , director of KU’s Center for Culturally Responsive Educational Neuroscience, emphasized the critical role of language in word problems, highlighting the importance of using culturally familiar terms. For instance, substituting "pastry" for "quesadilla" could significantly affect comprehension for students from diverse backgrounds. Failure to grasp the initial scenario can impede subsequent problem-solving efforts.

The study proved effective in improving students’ problem-solving abilities, despite covariates including an individual’s basic calculation skills, fluid intelligence and reading comprehension scores. That finding is key as, while ideally all students would begin on equal footing and there were little variations in a classroom, in reality, covariates exist and are commonplace.

The study had trained tutors deliver the intervention, and its effectiveness should be further tested with working teachers, the authors wrote. Orosco said professional development to help teachers gain the skills is necessary, and it is vital for teacher preparation programs to train future teachers with such skills as well. And helping students at the elementary level is necessary to help ensure success in future higher-level math classes such as algebra.

The research builds on Orosco and colleagues’ work in understanding and improving math instruction for English learners . Future work will continue to examine the role of cognitive functions such as working memory and brain science , as well as potential integration of artificial intelligence in teaching math.

“Comprehension strategy instruction helps students make connections, ask questions, visualize, synthesize and monitor their thinking about word problems,” Orosco and Reed wrote. “Finally, applying comprehension strategy instruction supports ELs in integrating their reading, language and math cognition… Focusing on relevant language in word problems and providing collaborative support significantly improved students’ solution accuracy.”

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A Proven Model to Combat U.S. Drug Shortages

Dan Liljenquist

By leaning into transparency, eschewing rebate schemes, and cutting out middlemen, this nonprofit is able to significantly lower the market price of insulin.

Drug shortages continue to plague the United States. In many ways, the problem is the result of deficiencies in the current pharma market. But a model for addressing this problem is showing that it isn’t intractable. The company employing this model is Civica Rx, which was established in 2018 by health systems and philanthropies to address shortages of generic sterile injectable drug. This article discusses the elements of its model and its achievements.

Drug shortages have been a chronic problem in the United States for well over a decade. A newly published report from the American Society of Health System Pharmacists (ASHP) shows drug shortages are at a record high of over 300 essential medications in short supply. Currently, on a national basis, we are seeing an acute exacerbation of shortages as several manufacturers experience quality problems, causing them to leave the market permanently or temporarily.

Dan Liljenquist is senior vice president and chief strategy officer at Intermountain Health and board chair of Civica Rx.

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Model Based Problem Solving Strategy for Simulation
An Overview Of 9 Step Problem Solving Model
Introduction to Problem Solving Skills
the 6 step problem solving model
Keith Glein Problem Solving Blog
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3.1: Check Your Understanding
SOLVING PROBLEM

COMMENTS

PDF THIRTEEN PROBLEM-SOLVING MODELS
The Six-Step method provides a focused procedure for the problem solving (PS) group. It ensures consistency, as everyone understands the approach to be used. By using data, it helps eliminate bias and preconceptions, leading to greater objectivity. It helps to remove divisions and encourages collaborative working.
Problem-Solving Models: What They Are and How To Use Them
Most problem-solving models rely on data to inform decisions, which helps to maintain objectivity and fairness throughout the process. By using problem-solving methods to hear the opinions of everyone, you can eliminate bias when solving a problem. In addition, implementing problem-solving models can lead to more effective, thoughtful solutions.
Common Problem-Solving Models & How to Use Them
The first step in creating a problem-solving plan is to understand what we mean when we say problem-solving models. A problem-solving model is a step-by-step process that helps a team identify and effectively solve problems that they may encounter. This problem-solving approach gives the team the muscle memory and guide to address a conflict ...
35 problem-solving techniques and methods for solving complex problems
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.
Mathematical modeling and problem solving: from fundamentals to
The rapidly advancing fields of machine learning and mathematical modeling, greatly enhanced by the recent growth in artificial intelligence, are the focus of this special issue. This issue compiles extensively revised and improved versions of the top papers from the workshop on Mathematical Modeling and Problem Solving at PDPTA'23, the 29th International Conference on Parallel and Distributed ...
The FOCUS Model
The model is helpful because it uses a team-based approach to problem solving and to business-process improvement, and this makes it particularly useful for solving cross-departmental process issues. Also, it encourages people to rely on objective data rather than on personal opinions, and this improves the quality of the outcome. It has five ...
Problem Posing and Modeling: Confronting the Dilemma of Rigor or
For both problem-posing and modeling, the result of this "collapse" can be that authentic processes (of problem solving or modeling) are caricatured, as the field sacrifices "relevance" in favor of "rigor"—both at the level of the mathematics (focusing exclusively on algebraic functions, one-way causal relations, etc.) and at the ...
Problem Solving Versus Modeling
Definitions of problem solving have been posed over the years, with no one definition emerging as the accepted one in the field. In the process of co-writing a chapter with Richard Lesh on mathematical problem solving and modeling for The Second Handbook of Research on Mathematics Teaching and Learning, I read a large number of definitions and discussions about the characteristics of problem ...
Foundations of a Models and Modeling Perspective on Mathematics
ABSTRACT. At the end of this chapter, Appendixes A, B, and C, are three examples of problem-solving activities that we refer to as model-eliciting activities—so called because the products that students produce go beyond short answers to narrowly specified questions—which involve sharable, manipulatable, modifiable, and reusable conceptual ...
Modeling Problem Solving
Modeling Problem Solving. We've discussed in previous chapters how part of a tutor's task is to model good learning habits. When tutors are organized, use good time management, and leverage resources, we demonstrate the skills that students can use to be successful learners. Problem-solving is an additional skill that tutors model for students.
PDF Creative Problem Solving
Problem Solving as the sum of its parts: Creative means having an element of newness and innovation, and relevance. Problem encompasses any situation that presents a challenge, offers an opportunity or is a concern. Solving means devising ways to answer, to meet or satisfy the problem. It can also mean adapting yourself to the situation or ...
The Problem-Solving Process
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 ...
The Six-Step Problem-Solving Model: A Collaborative Approach to
The Six Step Problem Solving Model isn't just a method; it's a mindset. A mindset that ensures problems are tackled systematically and collaboratively, driving teams towards effective ...
Using Mathematical Modeling to Get Real With Students
To solve a word problem, students can pick out the numbers and decide on an operation.". But through the use of mathematical modeling, students are plucked out of the hypothetical realm and plunged into the complexities of reality—presented with opportunities to help solve real-world problems with many variables by generating questions ...
What is 8D? Eight Disciplines Problem Solving Process
The 8D problem solving model establishes a permanent corrective action based on statistical analysis of the problem and focuses on the origin of the problem by determining its root causes. Although it originally comprised eight stages, or disciplines, the eight disciplines system was later augmented by an initial planning stage. ...
PDF Six-step Problem Solving Model
Using a problem solving model enables a group to consider all possible causes of a problem and all possible solutions. A problem solving model uses a series of logical steps to help a group identify the most important causes and the best solution. Following the model not only helps the group arrive at a solution, it helps the group arrive at a
The McKinsey guide to problem solving
The McKinsey guide to problem solving. Become a better problem solver with insights and advice from leaders around the world on topics including developing a problem-solving mindset, solving problems in uncertain times, problem solving with AI, and much more.
What is Problem Solving? Steps, Process & Techniques
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.
How to master the seven-step problem-solving process
Structured problem solving strategies can be used to address almost any complex challenge in business or public policy. ... Classic problem solving often gravitates toward a model; design thinking migrates toward a prototype. Rather than writing a big deck with all my supporting evidence, they'll bring an example, a thing, and that feels ...
Problem Solving & Modeling
17 Problem Solving & Modeling Introduction. Aerospace engineering students soon begin to ask when they can start to solve actual problems relevant to aircraft, rockets, or spacecraft. Of course, it is natural to ask such questions, even early on in an engineering education. However, practical problem-solving in engineering is a serious business ...
Research Projects
Through mixed methods research that includes group problem solving discussions, written reflection, and mock interviews, we are investigating the role of problem typology in helping students to: (i) recognize and orient themselves to different types of engineering problems; (ii) deconstruct and re-synthesize technical experiences in terms of ...
Problem Solving and Mathematical Modeling
Analyzing and classifying the problem is a central theme of problem solving. Essentially, looking into the skills require for software engineers includes technical skills like software designing, coding and software testing, problem-solving skills like logical and analytical thinking, problem modeling and the soft skills like communication, teamwork and negotiation.
[2404.18766] PECC: Problem Extraction and Coding Challenges
Recent advancements in large language models (LLMs) have showcased their exceptional abilities across various tasks, such as code generation, problem-solving and reasoning. Existing benchmarks evaluate tasks in isolation, yet the extent to which LLMs can understand prose-style tasks, identify the underlying problems, and then generate appropriate code solutions is still unexplored. Addressing ...
How to utilize problem-solving models in education
The MTSS problem-solving model is a data-driven decision-making process that helps educators utilize and analyze interventions based on students' needs on a continual basis. Traditionally, the MTSS problem-solving model only involves four steps: Identifying the student's strengths and needs, based on data.
Solon Schools' Future Problem Solving teams send 24 students to
SOLON, Ohio - Future Problem Solving teams from Solon High School, Solon Middle School and Orchard Middle School excelled at the State Bowl hosted by Solon recently. On Monday (April 29 ...
Enhancing Transformer Models with Filler Tokens: A Novel AI Approach to
Models trained with these tokens surpassed the baseline immediate-answer models and exhibited enhanced problem-solving abilities on more complex tasks. Specifically, filler tokens consistently improved model performance in setups where the input sequence involved higher-dimensional data, achieving accuracies near 100% on tasks that would ...
REBEL: A Reinforcement Learning RL Algorithm that Reduces the Problem
Initially designed for continuous control tasks, Proximal Policy Optimization (PPO) has become widely used in reinforcement learning (RL) applications, including fine-tuning generative models. However, PPO's effectiveness relies on multiple heuristics for stable convergence, such as value networks and clipping, making its implementation sensitive and complex. Despite this, RL demonstrates ...
Featured news and headlines
"Word problem-solving is influenced by both the science of reading and the science of math. Key components include number sense, decoding, language comprehension and working memory. Utilizing direct and explicit teaching methods enhances understanding and enables students to effectively connect these skills to solve math problems.
A Proven Model to Combat U.S. Drug Shortages
Drug shortages continue to plague the United States. In many ways, the problem is the result of deficiencies in the current pharma market. But a model for addressing this problem is showing that ...