problem solving through problems solutions manual

• © 1983

Problem-Solving Through Problems

• Loren C. Larson 0

Department of Mathematics, St. Olaf College, Northfield, USA

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Part of the book series: Problem Books in Mathematics (PBM)

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Table of contents (8 chapters)

Front matter.

Loren C. Larson

Two Important Principles: Induction and Pigeonhole

Summation of series, intermediate real analysis, inequalities, back matter.

• Cauchy-Schwarz Inequality
• Problem solving
• Schwarz inequality
• fundamental theorem
• mean value theorem
• pigeonhole principle

Book Title : Problem-Solving Through Problems

Authors : Loren C. Larson

Series Title : Problem Books in Mathematics

DOI : https://doi.org/10.1007/978-1-4612-5498-0

Publisher : Springer New York, NY

eBook Packages : Springer Book Archive

Copyright Information : Springer-Verlag New York Inc. 1983

Softcover ISBN : 978-0-387-96171-2 Published: 17 July 1985

eBook ISBN : 978-1-4612-5498-0 Published: 06 December 2012

Series ISSN : 0941-3502

Series E-ISSN : 2197-8506

Edition Number : 1

Number of Pages : XI, 352

Number of Illustrations : 11 b/w illustrations

Topics : Analysis , Mathematics, general

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Create an Effective Problem-Solving Manual at Work

Problems are an essential part of human existence. To grow, one must develop the capacity to solve problems. Companies must create teams with the skills to identify potential problems and find possible solutions. Understanding how to resolve challenges in the workplace increases employee and company value. This is where a problem-solving manual becomes an invaluable tool for productive work.

Problem-solving in an organisation enhances collaboration, which promotes collective thinking. This accounts for 80% of successful results from teams. Such collaboration engenders strong team development, excellent communication and a clear understanding of common challenges and potential solutions amongst teams.

What is a problem-solving manual?

Problem-solving is a technique that inspires parties in conflict to collaborate in defining problems, analysing their causes, proposing various solutions and selecting the best options for resolving the conflict.

The problem-solving manual should teach companies how to understand business challenges and the source of problems through a systematic process while providing clear direction on the best solution.

Benefits of a problem-solving manual

Structuring your problem-solving manual

The manual should contain a minimum five-step process for providing lasting solutions to the problems. These processes include:

• Conflict definition
• Analysis of the problem’s causes
• Brainstorming for resolution
• Selection of a preferred solution
• Implementation of the solution

Problem-solving is mostly applicable in small groups. It is important that people within a team learn analytical thinking. It must be made clear that agreeing on the causes of conflict is often difficult. The good thing about conflict management techniques is that they are positive approaches and appreciate existing capacities rather than the weaknesses of the teams.

Using a problem-solving manual for teamwork

Improves understanding.

One of the ways problem-solving assists teamwork is through the improvement of mutual understanding, thereby leading to an understanding of the context of the team’s operation and the responsibilities of the teammates. Collaborative techniques help members of the team appreciate each other’s efforts.

However, planning becomes difficult when team members work in shifts. TimeTrack Auto Scheduling provides a perfect overview for both management and the employees. Thus, managers can automatically personalise shifts based on projects and skills and employee availability.

TimeTrack Auto Scheduling

Enhances excellent communication

Working together in a team enhances the understanding of potential solutions and also creates opportunities for better communication. Each member brings a different context to the problem-solving scenario so good communication is a must for harmonious collaboration.

Effective handling of risks

A problem-solving manual equips a team to better anticipate risks. Because every team member comes with a different perspective, the team has more capacity to spot and mitigate risks in advance. A team must be willing to learn the requisite skills and have the confidence to accept the consequences of its choices.

Encourages smart working

A problem-solving manual streamlines processes to create a flexible work system that meets current needs. Smart working is the foundation of a seamless problem-solving environment, iterating processes to solve business challenges.

Highlights hidden potential

Compiling a problem-solving manual with the team’s input can uncover a hidden talent on the team. Or, you may realise that a certain employee has a natural aptitude for empathetic communication. When ideas and solutions for problem-solving flow, this broadens the potential for the company’s success. Keen problem-solving skills prepare team members to take on larger challenges and explore smart techniques for possible solutions.

Lower business failure

A problem-solving manual is essentially your back-up plan. All good leaders know that planning for failure is vital for any business. The problem-solving guide is a strategic preparation for learning opportunities, long-term growth and contingencies.

Low possibility of bias

In the same vein, each team member understands a fraction of the entire problem-solving process. The collective knowledge of the team fosters a better understanding of the problem and solution when the team engages in the search for the solution.

Tips to create an effective problem-solving manual

Define the problems.

Think big and think of any number of significant business challenges that your company could face. Review historical problems. Identify and clearly define the problem with substantial detail. The team saddled with devising problem-solving techniques must acknowledge the existence of the problem and, from divergent viewpoints, approach the issue with open-mindedness.

Evaluate the problems

After you have identified and named the problem, go into a process of reviews. This is process evaluation before suggesting a solution. To do this, analyse the issues from different perspectives for better understanding. To identify the root cause of your business problem, teams must ensure that the company’s approach is aligned with its goals.

Conduct a SWOT analysis

This approach involves pinpointing and acknowledging strengths , weaknesses , opportunities and threats to the company. This analysis allows businesses to become self-aware and identify challenges before they occur.

Conduct market research

Gathering and analysing data concerning your business should include regular customer surveys designed to gauge needs and reactions to your products. This helps identify customer pain points, buying habits and the demographics of the marketplace.

Gather team input

The input from a team will always be a substantial sticking point. Input from business advisors, your team or experts on the subject matter will unearth various ideas , solutions and challenges the organisation is facing. This is one of the most effective problem-solving techniques for long-term solutions.

Collate the problems and solutions

At this stage of the process, decide on the course of action to implement, based on the reviews and solutions people have suggested. This process involves weighing the pros and cons of the ideal solution. Take key actions after deciding on the best option.

Gauge the results of solutions

Measure the results of problem-solving techniques and processes by tracking progress. This will help gauge if the particular issue is resolved. Due to the complex nature of solving business challenges, you need to take baseline reviews, decide on key performance indicators and set goals and timelines.

Take responsibility

Whatever choice you make is a choice, and it should be superior to others. This means you must live with the consequences of your choice. Two outcomes await you in solving business challenges. Either you escalate the problem or you solve it.

The bottom line

The ultimate outcome of your problem-solving manual is to provide a blueprint of solutions to common workplace and business challenges.

Applying the wrong technique could result in a waste of time and resources. This is why proactiveness and diverse consultation is vital. Work with your team to create an efficient guide of solutions.

I am a researcher, writer, and self-published author. Over the last 9 years, I have dedicated my time to delivering unique content to startups and non-governmental organizations and have covered several topics, including wellness, technology, and entrepreneurship. I am now passionate about how time efficiency affects productivity, business performance, and profitability.

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Art of Problem Solving Introduction to Algebra Textbook and Solutions Manual 2-Book Set Paperback – January 1, 1965

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About the author

Richard rusczyk.

Richard Rusczyk founded Art of Problem Solving (AoPS) in 2003 to create interactive educational opportunities for avid math students. Richard is one of the co-authors of the Art of Problem Solving classic textbooks, author of Art of Problem Solving's Introduction to Algebra, Introduction to Geometry, and Precalculus textbooks, co-author of Art of Problem Solving's Intermediate Algebra and Prealgebra, one of the co-creators of the Mandelbrot Competition, and a past Director of the USA Mathematical Talent Search. He was a participant in National MATHCOUNTS, a three-time participant in the Math Olympiad Summer Program, and a USA Mathematical Olympiad winner (1989). He graduated from Princeton University in 1993, and worked as a bond trader for D.E. Shaw & Company for four years. AoPS marks Richard's return to his vocation: educating motivated students.

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

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

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

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

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

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

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

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

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

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

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

Key features of a good mathematics problem includes:

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

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

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

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

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

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

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

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

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

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

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

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

Number Talks

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

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

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

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

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

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

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

Instead, you and your students could say the following:

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

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

Using “Worksheets”

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

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

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

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

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

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

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

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

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

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

when we teach students how to problem solve

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

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

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

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

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10 Best Problem-Solving Therapy Worksheets & Activities

Cognitive science tells us that we regularly face not only well-defined problems but, importantly, many that are ill defined (Eysenck & Keane, 2015).

Sometimes, we find ourselves unable to overcome our daily problems or the inevitable (though hopefully infrequent) life traumas we face.

Problem-Solving Therapy aims to reduce the incidence and impact of mental health disorders and improve wellbeing by helping clients face life’s difficulties (Dobson, 2011).

This article introduces Problem-Solving Therapy and offers techniques, activities, and worksheets that mental health professionals can use with clients.

Before you continue, we thought you might like to download our three Positive Psychology Exercises for free . These science-based exercises explore fundamental aspects of positive psychology, including strengths, values, and self-compassion, and will give you the tools to enhance the wellbeing of your clients, students, or employees.

This Article Contains:

What is problem-solving therapy, 14 steps for problem-solving therapy, 3 best interventions and techniques, 7 activities and worksheets for your session, fascinating books on the topic, resources from positivepsychology.com, a take-home message.

Problem-Solving Therapy assumes that mental disorders arise in response to ineffective or maladaptive coping. By adopting a more realistic and optimistic view of coping, individuals can understand the role of emotions and develop actions to reduce distress and maintain mental wellbeing (Nezu & Nezu, 2009).

“Problem-solving therapy (PST) is a psychosocial intervention, generally considered to be under a cognitive-behavioral umbrella” (Nezu, Nezu, & D’Zurilla, 2013, p. ix). It aims to encourage the client to cope better with day-to-day problems and traumatic events and reduce their impact on mental and physical wellbeing.

Clinical research, counseling, and health psychology have shown PST to be highly effective in clients of all ages, ranging from children to the elderly, across multiple clinical settings, including schizophrenia, stress, and anxiety disorders (Dobson, 2011).

Can it help with depression?

PST appears particularly helpful in treating clients with depression. A recent analysis of 30 studies found that PST was an effective treatment with a similar degree of success as other successful therapies targeting depression (Cuijpers, Wit, Kleiboer, Karyotaki, & Ebert, 2020).

Other studies confirm the value of PST and its effectiveness at treating depression in multiple age groups and its capacity to combine with other therapies, including drug treatments (Dobson, 2011).

The major concepts

Effective coping varies depending on the situation, and treatment typically focuses on improving the environment and reducing emotional distress (Dobson, 2011).

PST is based on two overlapping models:

Social problem-solving model

This model focuses on solving the problem “as it occurs in the natural social environment,” combined with a general coping strategy and a method of self-control (Dobson, 2011, p. 198).

The model includes three central concepts:

• Social problem-solving
• The problem
• The solution

The model is a “self-directed cognitive-behavioral process by which an individual, couple, or group attempts to identify or discover effective solutions for specific problems encountered in everyday living” (Dobson, 2011, p. 199).

Relational problem-solving model

The theory of PST is underpinned by a relational problem-solving model, whereby stress is viewed in terms of the relationships between three factors:

• Stressful life events
• Emotional distress and wellbeing
• Problem-solving coping

Therefore, when a significant adverse life event occurs, it may require “sweeping readjustments in a person’s life” (Dobson, 2011, p. 202).

• Enhance positive problem orientation
• Decrease negative orientation
• Foster ability to apply rational problem-solving skills
• Reduce the tendency to avoid problem-solving
• Minimize the tendency to be careless and impulsive

D’Zurilla’s and Nezu’s model includes (modified from Dobson, 2011):

• Initial structuring Establish a positive therapeutic relationship that encourages optimism and explains the PST approach.
• Assessment Formally and informally assess areas of stress in the client’s life and their problem-solving strengths and weaknesses.
• Obstacles to effective problem-solving Explore typically human challenges to problem-solving, such as multitasking and the negative impact of stress. Introduce tools that can help, such as making lists, visualization, and breaking complex problems down.
• Problem orientation – fostering self-efficacy Introduce the importance of a positive problem orientation, adopting tools, such as visualization, to promote self-efficacy.
• Problem orientation – recognizing problems Help clients recognize issues as they occur and use problem checklists to ‘normalize’ the experience.
• Problem orientation – seeing problems as challenges Encourage clients to break free of harmful and restricted ways of thinking while learning how to argue from another point of view.
• Problem orientation – use and control emotions Help clients understand the role of emotions in problem-solving, including using feelings to inform the process and managing disruptive emotions (such as cognitive reframing and relaxation exercises).
• Problem orientation – stop and think Teach clients how to reduce impulsive and avoidance tendencies (visualizing a stop sign or traffic light).
• Problem definition and formulation Encourage an understanding of the nature of problems and set realistic goals and objectives.
• Generation of alternatives Work with clients to help them recognize the wide range of potential solutions to each problem (for example, brainstorming).
• Decision-making Encourage better decision-making through an improved understanding of the consequences of decisions and the value and likelihood of different outcomes.
• Solution implementation and verification Foster the client’s ability to carry out a solution plan, monitor its outcome, evaluate its effectiveness, and use self-reinforcement to increase the chance of success.
• Guided practice Encourage the application of problem-solving skills across multiple domains and future stressful problems.
• Rapid problem-solving Teach clients how to apply problem-solving questions and guidelines quickly in any given situation.

Success in PST depends on the effectiveness of its implementation; using the right approach is crucial (Dobson, 2011).

Problem-solving therapy – Baycrest

The following interventions and techniques are helpful when implementing more effective problem-solving approaches in client’s lives.

First, it is essential to consider if PST is the best approach for the client, based on the problems they present.

Is PPT appropriate?

It is vital to consider whether PST is appropriate for the client’s situation. Therapists new to the approach may require additional guidance (Nezu et al., 2013).

Therapists should consider the following questions before beginning PST with a client (modified from Nezu et al., 2013):

• Has PST proven effective in the past for the problem? For example, research has shown success with depression, generalized anxiety, back pain, Alzheimer’s disease, cancer, and supporting caregivers (Nezu et al., 2013).
• Is PST acceptable to the client?
• Is the individual experiencing a significant mental or physical health problem?

All affirmative answers suggest that PST would be a helpful technique to apply in this instance.

Five problem-solving steps

The following five steps are valuable when working with clients to help them cope with and manage their environment (modified from Dobson, 2011).

Ask the client to consider the following points (forming the acronym ADAPT) when confronted by a problem:

• Attitude Aim to adopt a positive, optimistic attitude to the problem and problem-solving process.
• Define Obtain all required facts and details of potential obstacles to define the problem.
• Alternatives Identify various alternative solutions and actions to overcome the obstacle and achieve the problem-solving goal.
• Predict Predict each alternative’s positive and negative outcomes and choose the one most likely to achieve the goal and maximize the benefits.
• Try out Once selected, try out the solution and monitor its effectiveness while engaging in self-reinforcement.

If the client is not satisfied with their solution, they can return to step ‘A’ and find a more appropriate solution.

Positive self-statements

When dealing with clients facing negative self-beliefs, it can be helpful for them to use positive self-statements.

Use the following (or add new) self-statements to replace harmful, negative thinking (modified from Dobson, 2011):

• I can solve this problem; I’ve tackled similar ones before.
• I can cope with this.
• I just need to take a breath and relax.
• Once I start, it will be easier.
• It’s okay to look out for myself.
• I can get help if needed.
• Other people feel the same way I do.
• I’ll take one piece of the problem at a time.
• I can keep my fears in check.
• I don’t need to please everyone.

5 Worksheets and workbooks

Problem-solving self-monitoring form.

Answering the questions in the Problem-Solving Self-Monitoring Form provides the therapist with necessary information regarding the client’s overall and specific problem-solving approaches and reactions (Dobson, 2011).

Ask the client to complete the following:

• Describe the problem you are facing.
• What is your goal?
• What have you tried so far to solve the problem?
• What was the outcome?

Reactions to Stress

It can be helpful for the client to recognize their own experiences of stress. Do they react angrily, withdraw, or give up (Dobson, 2011)?

The Reactions to Stress worksheet can be given to the client as homework to capture stressful events and their reactions. By recording how they felt, behaved, and thought, they can recognize repeating patterns.

What Are Your Unique Triggers?

Helping clients capture triggers for their stressful reactions can encourage emotional regulation.

When clients can identify triggers that may lead to a negative response, they can stop the experience or slow down their emotional reaction (Dobson, 2011).

The What Are Your Unique Triggers ? worksheet helps the client identify their triggers (e.g., conflict, relationships, physical environment, etc.).

Problem-Solving worksheet

Imagining an existing or potential problem and working through how to resolve it can be a powerful exercise for the client.

Use the Problem-Solving worksheet to state a problem and goal and consider the obstacles in the way. Then explore options for achieving the goal, along with their pros and cons, to assess the best action plan.

Getting the Facts

Clients can become better equipped to tackle problems and choose the right course of action by recognizing facts versus assumptions and gathering all the necessary information (Dobson, 2011).

Use the Getting the Facts worksheet to answer the following questions clearly and unambiguously:

• Who is involved?
• What did or did not happen, and how did it bother you?
• Where did it happen?
• When did it happen?
• Why did it happen?
• How did you respond?

2 Helpful Group Activities

While therapists can use the worksheets above in group situations, the following two interventions work particularly well with more than one person.

Generating Alternative Solutions and Better Decision-Making

A group setting can provide an ideal opportunity to share a problem and identify potential solutions arising from multiple perspectives.

Use the Generating Alternative Solutions and Better Decision-Making worksheet and ask the client to explain the situation or problem to the group and the obstacles in the way.

Once the approaches are captured and reviewed, the individual can share their decision-making process with the group if they want further feedback.

Visualization

Visualization can be performed with individuals or in a group setting to help clients solve problems in multiple ways, including (Dobson, 2011):

• Clarifying the problem by looking at it from multiple perspectives
• Rehearsing a solution in the mind to improve and get more practice
• Visualizing a ‘safe place’ for relaxation, slowing down, and stress management

Guided imagery is particularly valuable for encouraging the group to take a ‘mental vacation’ and let go of stress.

Ask the group to begin with slow, deep breathing that fills the entire diaphragm. Then ask them to visualize a favorite scene (real or imagined) that makes them feel relaxed, perhaps beside a gently flowing river, a summer meadow, or at the beach.

The more the senses are engaged, the more real the experience. Ask the group to think about what they can hear, see, touch, smell, and even taste.

Encourage them to experience the situation as fully as possible, immersing themselves and enjoying their place of safety.

Such feelings of relaxation may be able to help clients fall asleep, relieve stress, and become more ready to solve problems.

We have included three of our favorite books on the subject of Problem-Solving Therapy below.

1. Problem-Solving Therapy: A Treatment Manual – Arthur Nezu, Christine Maguth Nezu, and Thomas D’Zurilla

This is an incredibly valuable book for anyone wishing to understand the principles and practice behind PST.

Written by the co-developers of PST, the manual provides powerful toolkits to overcome cognitive overload, emotional dysregulation, and the barriers to practical problem-solving.

Find the book on Amazon .

2. Emotion-Centered Problem-Solving Therapy: Treatment Guidelines – Arthur Nezu and Christine Maguth Nezu

Another, more recent, book from the creators of PST, this text includes important advances in neuroscience underpinning the role of emotion in behavioral treatment.

Along with clinical examples, the book also includes crucial toolkits that form part of a stepped model for the application of PST.

3. Handbook of Cognitive-Behavioral Therapies – Keith Dobson and David Dozois

This is the fourth edition of a hugely popular guide to Cognitive-Behavioral Therapies and includes a valuable and insightful section on Problem-Solving Therapy.

This is an important book for students and more experienced therapists wishing to form a high-level and in-depth understanding of the tools and techniques available to Cognitive-Behavioral Therapists.

For even more tools to help strengthen your clients’ problem-solving skills, check out the following free worksheets from our blog.

• Case Formulation Worksheet This worksheet presents a four-step framework to help therapists and their clients come to a shared understanding of the client’s presenting problem.
• Understanding Your Default Problem-Solving Approach This worksheet poses a series of questions helping clients reflect on their typical cognitive, emotional, and behavioral responses to problems.
• Social Problem Solving: Step by Step This worksheet presents a streamlined template to help clients define a problem, generate possible courses of action, and evaluate the effectiveness of an implemented solution.
• 17 Positive Psychology Exercises If you’re looking for more science-based ways to help others enhance their wellbeing, check out this signature collection of 17 validated positive psychology tools for practitioners . Use them to help others flourish and thrive.

While we are born problem-solvers, facing an incredibly diverse set of challenges daily, we sometimes need support.

Problem-Solving Therapy aims to reduce stress and associated mental health disorders and improve wellbeing by improving our ability to cope. PST is valuable in diverse clinical settings, ranging from depression to schizophrenia, with research suggesting it as a highly effective treatment for teaching coping strategies and reducing emotional distress.

Many PST techniques are available to help improve clients’ positive outlook on obstacles while reducing avoidance of problem situations and the tendency to be careless and impulsive.

The PST model typically assesses the client’s strengths, weaknesses, and coping strategies when facing problems before encouraging a healthy experience of and relationship with problem-solving.

Why not use this article to explore the theory behind PST and try out some of our powerful tools and interventions with your clients to help them with their decision-making, coping, and problem-solving?

We hope you enjoyed reading this article. Don’t forget to download our three Positive Psychology Exercises for free .

• Cuijpers, P., Wit, L., Kleiboer, A., Karyotaki, E., & Ebert, D. (2020). Problem-solving therapy for adult depression: An updated meta-analysis. European P sychiatry ,  48 (1), 27–37.
• Dobson, K. S. (2011). Handbook of cognitive-behavioral therapies (3rd ed.). Guilford Press.
• Dobson, K. S., & Dozois, D. J. A. (2021). Handbook of cognitive-behavioral therapies  (4th ed.). Guilford Press.
• Eysenck, M. W., & Keane, M. T. (2015). Cognitive psychology: A student’s handbook . Psychology Press.
• Nezu, A. M., & Nezu, C. M. (2009). Problem-solving therapy DVD . Retrieved September 13, 2021, from https://www.apa.org/pubs/videos/4310852
• Nezu, A. M., & Nezu, C. M. (2018). Emotion-centered problem-solving therapy: Treatment guidelines. Springer.
• Nezu, A. M., Nezu, C. M., & D’Zurilla, T. J. (2013). Problem-solving therapy: A treatment manual . Springer.

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