solving life's problems pdf

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Solving Life's Problems

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7 Solution-Focused Therapy Techniques and Worksheets (+PDF)

It has analyzed a person’s problems from where they started and how those problems have an effect on that person’s life.

Out of years of observation of family therapy sessions, the theory and applications of solution-focused therapy developed.

Let’s explore the therapy, along with techniques and applications of the approach.

Before you read on, we thought you might like to download our three Positive Psychology Exercises for free . These science-based exercises will 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:

5 solution-focused therapy techniques, handy sft worksheets (pdf), solution-focused therapy interventions, 5 sft questions to ask clients, solution-focused brief therapy (sfbt techniques), 4 activities & exercises, best sft books, a take-home message.

Solution-focused therapy is a type of treatment that highlights a client’s ability to solve problems, rather than why or how the problem was created. It was developed over some time after observations of therapists in a mental health facility in Wisconsin by Steve de Shazer and Insoo Kim Berg and their colleagues.

Like positive psychology, Solution Focused Therapy (SFT) practitioners focus on goal-oriented questioning to assist a client in moving into a future-oriented direction.

Solution-focused therapy has been successfully applied to a wide variety of client concerns due to its broad application. It has been utilized in a wide variety of client groups as well. The approach presupposes that clients have some knowledge of what will improve their lives.

The following areas have utilized SFT with varying success:

• relationship difficulties
• drug and alcohol abuse
• eating disorders
• anger management
• communication difficulties
• crisis intervention
• incarceration recidivism reduction

Goal clarification is an important technique in SFT. A therapist will need to guide a client to envision a future without the problem with which they presented. With coaching and positive questioning, this vision becomes much more clarified.

With any presenting client concern, the main technique in SFT is illuminating the exception. The therapist will guide the client to an area of their life where there is an exception to the problem. The exception is where things worked well, despite the problem. Within the exception, an approach for a solution may be forged.

The ‘miracle question’ is another technique frequently used in SFT. It is a powerful tool that helps clients to move into a solution orientation. This question allows clients to begin small steps toward finding solutions to presenting problems (Santa Rita Jr., 1998). It is asked in a specific way and is outlined later in this article.

Experiment invitation is another way that therapists guide clients into solution orientation. By inviting clients to build on what is already working, clients automatically focus on the positive. In positive psychology, we know that this allows the client’s mind to broaden and build from that orientation.

Utilizing what has been working experimentally allows the client to find what does and doesn’t work in solving the issue at hand. During the second half of a consultation with a client, many SFT therapists take a break to reflect on what they’ve learned during the beginning of the session.

Consultation breaks and invitations for more information from clients allow for both the therapist and client to brainstorm on what might have been missed during the initial conversations. After this break, clients are complemented and given a therapeutic message about the presenting issue. The message is typically stated in the positive so that clients leave with a positive orientation toward their goals.

Here are four handy worksheets for use with solution-focused therapy.

• Miracle worksheet
• Exceptions to the Problem Worksheet
• Scaling Questions Worksheet
• SMART+ Goals Worksheet

Download 3 Free Positive Psychology Exercises (PDF)

Enhance wellbeing with these free, science-based exercises that draw on the latest insights from positive psychology.

Download 3 Free Positive Psychology Tools Pack (PDF)

By filling out your name and email address below.

Compliments are frequently used in SFT, to help the client begin to focus on what is working, rather than what is not. Acknowledging that a client has an impact on the movement toward a goal allows hope to become present. Once hope and perspective shift occurs, a client can decide what daily actions they would like to take in attaining a goal.

Higher levels of hope and optimism can predict the following desirable outcomes (Peterson & Seligman, 2004):

• achievement in all sorts of areas
• freedom from anxiety and depression
• improved social relationships
• improved physical well being

Mind mapping is an effective intervention also used to increase hope and optimism. This intervention is often used in life coaching practices. A research study done on solution-focused life coaching (Green, Oades, & Grant, 2006) showed that this type of intervention increases goal striving and hope, in addition to overall well-being.

Though life coaching is not the same as therapy, this study shows the effectiveness of improving positive behavior through solution-focused questioning.

Mind mapping is a visual thinking tool that helps structure information. It helps clients to better analyze, comprehend, and generate new ideas in areas they might not have been automatically self-generated. Having it on paper gives them a reference point for future goal setting as well.

Empathy is vital in the administration of SFBT. A client needs to feel heard and held by the practitioner for any forward movement to occur. Intentionally leaning in to ensure that a client knows that the practitioner is engaged in listening is recommended.

Speaking to strengths and aligning those strengths with goal setting are important interventions in SFT. Recognizing and acknowledging what is already working for the client validates strengths. Self-recognition of these strengths increases self-esteem and in turn, improves forward movement.

The questions asked in Solution-Focused Therapy are positively directed and in a goal-oriented stance. The intention is to allow a perspective shift by guiding clients in the direction of hope and optimism to lead them to a path of positive change. Results and progress come from focusing on the changes that need to be made for goal attainment and increased well being.

1. Miracle Question

Here is a clear example of how to administer the miracle question. It should be delivered deliberately. When done so, it allows the client to imagine the miracle occurring.

“ Now, I want to ask you a strange question. Suppose that while you are sleeping tonight and the entire house is quiet, a miracle happens. The miracle is that the problem which brought you here is solved. However, because you are sleeping, you don’t know that the miracle has happened. So, when you wake up tomorrow morning, what will be different that will tell you that a miracle has happened and the problem which brought you here is solved? ” (de Shazer, 1988)

2. Presupposing change questions

A practitioner of solution-focused therapy asks questions in an approach derived way.

Here are a few examples of presupposing change questions:

“What stopped complete disaster from occurring?” “How did you avoid falling apart.” “What kept you from unraveling?”

3. Exception Questions

Examples of exception questions include:

1. Tell me about times when you don’t get angry. 2. Tell me about times you felt the happiest. 3. When was the last time that you feel you had a better day? 4. Was there ever a time when you felt happy in your relationship? 5. What was it about that day that made it a better day? 6. Can you think of a time when the problem was not present in your life?

4. Scaling Questions

These are questions that allow a client to rate their experience. They also allow for a client to evaluate their motivation to change their experience. Scaling questions allow for a practitioner to add a follow-up question that is in the positive as well.

An example of a scaling question: “On a scale of 1-10, with 10 representing the best it can be and one the worst, where would you say you are today?”

A follow-up question: “ Why a four and not a five?”

Questions like these allow the client to explore the positive, as well as their commitment to the changes that need to occur.

5. Coping Questions

These types of questions open clients up to their resiliency. Clients are experts in their life experience. Helping them see what works, allows them to grow from a place of strength.

“How have you managed so far?” “What have you done to stay afloat?” “What is working?”

3 Scaling questions from Solution Focused Therapy – Uncommon Practitioners

The main idea behind SFBT is that the techniques are positively and solution-focused to allow a brief amount of time for the client to be in therapy. Overall, improving the quality of life for each client, with them at the center and in the driver’s seat of their growth. SFBT typically has an average of 5-8 sessions.

During the sessions, goals are set. Specific experimental actions are explored and deployed into the client’s daily life. By keeping track of what works and where adjustments need to be made, a client is better able to track his or her progress.

A method has developed from the Miracle Question entitled, The Miracle Method . The steps follow below (Miller & Berg, 1996). It was designed for combatting problematic drinking but is useful in all areas of change.

• State your desire for something in your life to be different.
• Envision a miracle happening, and your life IS different.
• Make sure the miracle is important to you.
• Keep the miracle small.
• Define the change with language that is positive, specific, and behavioral.
• State how you will start your journey, rather than how you will end it.
• Be clear about who, where, and when, but not the why.

A short selection of exercises which can be used

1. Solution-focused art therapy/ letter writing

A powerful in-session task is to request a client to draw or write about one of the following, as part of art therapy :

• a picture of their miracle
• something the client does well
• a day when everything went well. What was different about that day?
• a special person in their life

2. Strengths Finders

Have a client focus on a time when they felt their strongest. Ask them to highlight what strengths were present when things were going well. This can be an illuminating activity that helps clients focus on the strengths they already have inside of them.

A variation of this task is to have a client ask people who are important in their lives to tell them how they view the client’s strengths. Collecting strengths from another’s perspective can be very illuminating and helpful in bringing a client into a strength perspective.

3. Solution Mind Mapping

A creative way to guide a client into a brainstorm of solutions is by mind mapping. Have the miracle at the center of the mind map. From the center, have a client create branches of solutions to make that miracle happen. By exploring solution options, a client will self-generate and be more connected to the outcome.

4. Experiment Journals

Encourage clients to do experiments in real-life settings concerning the presenting problem. Have the client keep track of what works from an approach perspective. Reassure the client that a variety of experiments is a helpful approach.

17 Top-Rated Positive Psychology Exercises for Practitioners

Expand your arsenal and impact with these 17 Positive Psychology Exercises [PDF] , scientifically designed to promote human flourishing, meaning, and wellbeing.

Created by Experts. 100% Science-based.

These books are recommended reads for solution-focused therapy.

1. The Miracle Method: A Radically New Approach to Problem Drinking – Insoo Kim Berg and Scott D. Miller Ph.D.

The Miracle Method by Scott D. Miller and Insoo Kim Berg is a book that has helped many clients overcome problematic drinking since the 1990s.

By utilizing the miracle question in the book, those with problematic drinking behaviors are given the ability to envision a future without the problem.

Concrete, obtainable steps in reaching the envisioned future are laid out in this supportive read.

Available on Amazon .

2. Solution Focused Brief Therapy: 100 Key Points and Techniques – Harvey Ratney, Evan George and Chris Iveson

Solution Focused Brief Therapy: 100 Key Points and Techniques is a well-received book on solution-focused therapy. Authors Ratner, George, and Iveson provide a concisely written and easily understandable guide to the approach.

Its accessibility allows for quick and effective change in people’s lives.

The book covers the approach’s history, philosophical underpinnings, techniques, and applications. It can be utilized in organizations, coaching, leadership, school-based work, and even in families.

The work is useful for any practitioner seeking to learn the approach and bring it into practice.

3. Handbook of Solution-Focused Brief Therapy (Jossey-Bass Psychology) – Scott D. Miller, Mark Hubble and Barry L. Duncan

It includes work from 28 of the lead practitioners in the field and how they have integrated the solution-focused approach with the problem-focused approach.

It utilizes research across treatment modalities to better equip new practitioners with as many tools as possible.

4. More Than Miracles: The State of the Art of Solution -Focused Therapy  (Routledge Mental Health Classic Editions) – Steve de Shazer and Yvonne Dolan

It allows the reader to peek into hundreds of hours of observation of psychotherapy.

It highlights what questions work and provides a thoughtful overview of applications to complex problems.

Solution-Focused Therapy is an approach that empowers clients to own their abilities in solving life’s problems. Rather than traditional psychotherapy that focuses on how a problem was derived, SFT allows for a goal-oriented focus to problem-solving. This approach allows for future-oriented, rather than past-oriented discussions to move a client forward toward the resolutions of their present problem.

This approach is used in many different areas, including education, family therapy , and even in office settings. Creating cooperative and collaborative opportunities to problem solve allows mind-broadening capabilities. Illuminating a path of choice is a compelling way to enable people to explore how exactly they want to show up in this world.

Thanks for reading!

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

• de Shazer, S. (1988). Clues: Investigating solutions in brief therapy. New York, NY: W.W. Norton and Co.
• Green, L. S., Oades, L. G., & Grant, A. M. (2006). Cognitive-behavioral, solution-focused life coaching: Enhancing goal striving, well-being, and hope. The Journal of Positive Psychology, 1 (3), 142-149.
• Miller, S. D., & Berg, I. K. (1996). The miracle method: A radically new approach to problem drinking. New York, NY: W.W. Norton and Co.
• Peterson, C., & Seligman, M. E. P., (2004).  Character strengths and virtues: A handbook and classification (Vol. 1). New York, NY: Oxford University Press.
• Santa Rita Jr, E. (1998). What do you do after asking the miracle question in solution-focused therapy. Family Therapy, 25( 3), 189-195.

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Navigating Real-Life Situations: Social Problem Solving Scenarios PDF

As we navigate through life, we encounter various social situations that require problem-solving skills. Whether it’s resolving conflicts, making decisions, or understanding others’ perspectives, social problem-solving skills play a crucial role in our daily interactions. In this blog post, we will explore the importance of social problem-solving skills and how social problem solving scenarios PDFs can be a valuable tool in developing these skills.

Understanding Social Problem Solving

Social problem solving refers to the process of identifying, generating, evaluating, and implementing solutions to social challenges. It involves a series of steps that help individuals navigate real-life situations effectively. Let’s break down the components of social problem solving:

• Identifying the problem: The first step is to recognize and define the problem at hand. This requires individuals to accurately perceive the situation and understand the underlying issues.
• Generating possible solutions: Once the problem is identified, individuals need to brainstorm potential solutions. This step encourages creativity and critical thinking.
• Evaluating and selecting the best solution: After generating multiple solutions, individuals must evaluate each option based on its feasibility and potential outcomes. The goal is to select the most effective solution.
• Implementing the chosen solution: Once a solution is chosen, individuals need to put it into action. This step involves effective communication, collaboration, and decision-making.
• Reflecting on the outcome: Finally, individuals should reflect on the outcome of their chosen solution. This reflection helps them learn from their experiences and make adjustments for future situations.

Benefits of Social Problem Solving Scenarios

Engaging in social problem solving scenarios offers numerous benefits, both for individuals and their social-emotional development. Let’s explore some of these benefits:

• Enhancing critical thinking skills: Social problem solving scenarios require individuals to think critically and analyze various aspects of a situation. This process helps develop their ability to make informed decisions and solve complex problems.
• Developing empathy and perspective-taking abilities: By engaging with different scenarios, individuals can gain a deeper understanding of others’ perspectives and develop empathy. This skill is essential for building positive relationships and resolving conflicts.
• Improving communication and collaboration skills: Social problem solving scenarios often involve working with others to find solutions. This collaborative process helps individuals improve their communication, teamwork, and negotiation skills.

Introducing Social Problem Solving Scenarios PDF

A social problem solving scenarios PDF is a collection of real-life situations that individuals can use to practice their problem-solving skills. These scenarios are designed to simulate common social challenges and provide individuals with opportunities to navigate them effectively. They often include prompts, questions, and suggested solutions to guide the problem-solving process.

These PDFs can be used in various settings, such as classrooms, therapy sessions, or even at home. They are suitable for individuals of all ages and can be tailored to specific social-emotional learning goals.

There are several resources available online where you can find social problem solving scenarios PDFs. Websites like EverydaySpeech offer a wide range of scenarios that cover various social situations. These resources are often categorized based on age group, skill level, or specific topics, making it easier to find scenarios that suit your needs.

Tips for Navigating Real-Life Situations Using Social Problem Solving Scenarios PDF

Now that you have access to social problem solving scenarios PDFs, here are some tips to help you effectively navigate real-life situations:

• Familiarize yourself with the scenarios: Take the time to read and understand the scenarios before engaging in problem-solving discussions. This will ensure that you are well-prepared and can guide others through the process.
• Encourage active participation and discussion: Create a safe and inclusive environment where individuals feel comfortable sharing their thoughts and ideas. Encourage active participation and facilitate meaningful discussions to explore different perspectives.
• Guide individuals through the problem-solving process: Use the steps of social problem solving (identifying the problem, generating solutions, evaluating and selecting the best solution, implementing the chosen solution, and reflecting on the outcome) as a framework to guide individuals through the scenarios. Ask open-ended questions to stimulate critical thinking.
• Foster reflection and learning from the scenarios: After navigating a scenario, encourage individuals to reflect on the outcomes and discuss what they have learned. This reflection helps consolidate their problem-solving skills and prepares them for future social challenges.

Social problem-solving skills are essential for navigating real-life situations effectively. By engaging in social problem solving scenarios, individuals can enhance their critical thinking, empathy, communication, and collaboration skills. Social problem solving scenarios PDFs provide a practical and accessible tool for developing these skills.

So, why not start incorporating social problem solving scenarios PDFs into your social-emotional learning journey? Take the first step by accessing a free trial of EverydaySpeech’s resources. Start your EverydaySpeech free trial today and unlock a world of social problem-solving opportunities!

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SCAMPER Technique: Real Examples & Free Templates in Word, PDF, PPT…

Nathan Brooks

Published on May 24, 2024, updated on May 27, 2024

The SCAMPER technique is an invaluable tool designed to ignite innovative thoughts, offering a robust approach to problem-solving. In this guide, we'll delve into the world of SCAMPER, providing real-life examples and free templates in various formats including Word, PDF, PPT, and more. Ready to explore? Keep reading!

What is the SCAMPER Model?

The SCAMPER model is a valuable tool for prompting creative thinking and problem-solving. The acronym stands for:

Substitute: What elements of the product or process can you replace with something else?

Combine: How can you combine the product or process with something else to make it better?

Adapt: What can you change about the product or process to improve it?

Modify: Can you modify or change the product or process in any way?

Put to Another Use: Can you use the product or process in a different way or a different setting?

Eliminate: What parts of the product or process can you remove or simplify?

Reverse/Rearrange: Can you change the order of the product or process? Or reverse its components?

Each letter of SCAMPER represents a different strategy that prompts questions aimed at innovating and improving a product, process, or service. It's a practical method that encourages us to think about a problem from different perspectives, leading to fresh insights and novel solutions.

Real Examples of SCAMPER

SCAMPER is a creative thinking technique used to spark innovation and generate new ideas by asking specific questions about existing products or processes.

SCAMPER Example: Smartphone Evolution

Let's take a real-world example: the evolution of the smartphone.

1. Substitute:

   - Original: Physical keyboards on smartphones (like BlackBerry).

   - New: Touchscreen keyboards (as popularized by the iPhone).

   - Effect: This substitution allowed for larger screens and a more versatile user interface, making smartphones more user-friendly and capable of supporting a wider range of applications.

2. Combine:

   - Original: Separate devices for phone calls, cameras, and music players.

   - New: Integrated devices that combine calling, photography, and music capabilities.

   - Effect: This combination has led to the development of multifunctional smartphones that can replace several standalone devices, increasing convenience and portability.

   - Original: Personal digital assistants (PDAs) with limited internet connectivity.

   - New: Smartphones with full internet capabilities, adapting features from computers.

   - Effect: Adapting computer-like internet capabilities into smartphones enabled users to browse the web, use email, and access a wide range of online services on the go.

   - Original: Smartphones with fixed screen sizes.

   - New: Smartphones with flexible and foldable screens.

   - Effect: Modifying screen technology to create foldable phones has led to devices that can switch between a compact form and a larger display, offering enhanced versatility and user experience.

5. Put to another use:

   - Original: Smartphones primarily for communication.

   - New: Smartphones as payment devices using NFC technology.

   - Effect: This new use has transformed smartphones into convenient tools for cashless transactions, making them integral to modern financial practices.

6. Eliminate:

   - Original: Headphone jacks on smartphones.

   - New: Removal of headphone jacks in favor of wireless earbuds.

   - Effect: Eliminating the headphone jack has pushed the market towards wireless audio solutions, enhancing convenience and paving the way for advancements in Bluetooth technology.

7. Rearrange:

   - Original: Fixed camera positions on the back of the phone.

   - New: Pop-up, rotating, or under-display cameras.

   - Effect: Rearranging the camera positions has allowed for innovative design solutions that maximize screen real estate, creating a more immersive viewing experience and maintaining camera functionality.

Using the SCAMPER technique, companies have been able to continuously innovate and improve the smartphone, making it an indispensable part of modern life.

SCAMPER Technique in Writing

The SCAMPER technique, initially developed as a tool for product innovation, can also serve as a powerful ally in enhancing the creativity and effectiveness of your writing. Each of its seven strategies can be used to reframe your thinking and explore fresh perspectives. Let's delve into how each strategy applies to writing:

1. Substitute

Consider replacing characters, settings, or events with alternatives that could add depth or create an unexpected twist. Substituting words and phrases can also change the tone and style of your writing.

Merge different stories or ideas to create something new and interesting. You can also combine different writing styles, genres, or perspectives to generate innovative narratives.

Modify your writing style or voice to suit different audiences or contexts. Consider adapting successful structures or themes from other works and fitting them into your piece.

Alter aspects of your story, such as plot developments, character arcs, or conflict resolutions, to enhance engagement and intrigue. Modifying sentence structures and vocabulary can also enrich your text's stylistic range.

Put to Another Use

Reframe your story in a different context. A dramatic narrative can be reimagined as a comedic piece or a contemporary setting can shift to a historical one.

5. Eliminate

Remove unnecessary details, characters, or subplots that do not contribute to the main story. In the realm of language usage, eliminate redundancies and overly complex structures to enhance clarity.

6. Reverse/Rearrange

Change the order of events in your story, adopt a non-linear timeline, or tell the story from the end. Rearranging paragraphs and sentences can also improve the flow and coherence of your writing.

The SCAMPER technique is a dynamic approach to developing more creative and compelling content. Challenging conventional thinking and exploring diverse possibilities, it stimulates originality and inventiveness in your writing. So why not give it a try? The next time you're faced with writer's block or feel stuck in a creativity rut, remember SCAMPER and unleash your inner writer's full potential!

Boardmix: Your Free Gateway to SCAMPER Tool and SCAMPER Templates

Looking for a versatile tool that can enhance your creative thinking and problem-solving abilities? Boardmix brings you the perfect solution with its free SCAMPER tool and  SCAMPER templates .

Advantages of Using Boardmix as A SCAMPER Tool

Whether you're a student, entrepreneur, product manager, or designer, Boardmix serves as a powerful ally to foster your creative potential. It allows you to:

- Reframe your established thinking patterns

- Generate out-of-the-box ideas

- Enhance problem-solving abilities

- Improve decision-making processes

By offering this dynamic tool for free, Boardmix ensures that everyone can access this method to enhance their creativity without worrying about the cost.

A Look at the SCAMPER Templates on Boardmix

To help users better navigate through the SCAMPER process, Boardmix provides a selection of ready-to-use SCAMPER templates. These templates provide a structured approach to applying the SCAMPER technique, saving time and energy that can be channeled into the creative process.

Using the templates is simple. Just navigate to the Template library, search for the SCAMPER template, and use it immediately! Each one is divided into sections corresponding to each of the SCAMPER strategies. Users can fill in their ideas and insights in the provided spaces. This makes it easy to see how different thoughts connect and interact under each strategy, promoting a holistic understanding of the issue at hand.

Export and Convert SCAMPER Templates in Various Formats

Besides the free template to use online, one of the remarkable features of Boardmix is its ability to export SCAMPER templates into various formats such as Word, PDF, PPT, Image, and more.

SCAMPER Template in Word

Boardmix's ability to convert your SCAMPER template to a Word document offers a convenient way to create and manage your ideas. Microsoft Word's vast array of editing and formatting tools allows you to organize your thoughts efficiently under each SCAMPER category. Sharing and collaboration become effortless as Word files are universally accepted and easily accessible.

SCAMPER Template in PDF

If you're looking for a stable format that maintains the same appearance across different devices and operating systems, the PDF option provided by Boardmix is the perfect fit. The SCAMPER template in PDF format retains the original layout, providing a clear and concise view of your innovative ideas. Furthermore, PDF files can be secured with password protection, ensuring your data's safety and integrity.

SCAMPER Template in PPT

For those who wish to present their SCAMPER analysis creatively and visually, Boardmix enables conversion to PowerPoint (PPT)format. Just select your SCAMPER content and use the Boardmix AI to convert it easily! It's an ideal choice when discussing concepts with your team or during group brainstorming sessions.

SCAMPER Template as Image

Boardmix even allows you to convert your SCAMPER templates into image files. This can be particularly helpful if you want a quick visual representation of your ideas or if you need to insert the SCAMPER template into another document or presentation. Whichever format you choose, Boardmix ensures that your path to innovation remains smooth and hassle-free.

With Boardmix's powerful AI capabilities and intuitive interface, navigating through the SCAMPER technique becomes a seamless experience. The platform also enables real-time collaboration, making it easier for teams to brainstorm together, no matter where they are.

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A Better Framework for Solving Tough Problems

Start with trust and end with speed.

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When it comes to solving complicated problems, the default for many organizational leaders is to take their time to work through the issues at hand. Unfortunately, that often leads to patchwork solutions or problems not truly getting resolved.

But Anne Morriss offers a different framework. In this episode, she outlines a five-step process for solving any problem and explains why starting with trust and ending with speed is so important for effective change leadership. As she says, “Let’s get into dialogue with the people who are also impacted by the problem before we start running down the path of solving it.”

Morriss is an entrepreneur and leadership coach. She’s also the coauthor of the book, Move Fast and Fix Things: The Trusted Leader’s Guide to Solving Hard Problems .

Key episode topics include: strategy, decision making and problem solving, strategy execution, managing people, collaboration and teams, trustworthiness, organizational culture, change leadership, problem solving, leadership.

HBR On Strategy curates the best case studies and conversations with the world’s top business and management experts, to help you unlock new ways of doing business. New episodes every week.

• Listen to the full HBR IdeaCast episode: How to Solve Tough Problems Better and Faster (2023)
• Find more episodes of HBR IdeaCast
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HANNAH BATES: Welcome to HBR On Strategy , case studies and conversations with the world’s top business and management experts, hand-selected to help you unlock new ways of doing business.

When it comes to solving complicated problems, many leaders only focus on the most apparent issues. Unfortunately that often leads to patchwork or partial solutions. But Anne Morriss offers a different framework that aims to truly tackle big problems by first leaning into trust and then focusing on speed.

Morriss is an entrepreneur and leadership coach. She’s also the co-author of the book, Move Fast and Fix Things: The Trusted Leader’s Guide to Solving Hard Problems . In this episode, she outlines a five-step process for solving any problem. Some, she says, can be solved in a week, while others take much longer. She also explains why starting with trust and ending with speed is so important for effective change leadership.

This episode originally aired on HBR IdeaCast in October 2023. Here it is.

CURT NICKISCH: Welcome to the HBR IdeaCast from Harvard Business Review. I’m Curt Nickisch.

Problems can be intimidating. Sure, some problems are fun to dig into. You roll up your sleeves, you just take care of them; but others, well, they’re complicated. Sometimes it’s hard to wrap your brain around a problem, much less fix it.

And that’s especially true for leaders in organizations where problems are often layered and complex. They sometimes demand technical, financial, or interpersonal knowledge to fix. And whether it’s avoidance on the leaders’ part or just the perception that a problem is systemic or even intractable, problems find a way to endure, to keep going, to keep being a problem that everyone tries to work around or just puts up with.

But today’s guest says that just compounds it and makes the problem harder to fix. Instead, she says, speed and momentum are key to overcoming a problem.

Anne Morriss is an entrepreneur, leadership coach and founder of the Leadership Consortium and with Harvard Business School Professor Francis Frei, she wrote the new book, Move Fast and Fix Things: The Trusted Leaders Guide to Solving Hard Problems . Anne, welcome back to the show.

ANNE MORRISS: Curt, thank you so much for having me.

CURT NICKISCH: So, to generate momentum at an organization, you say that you really need speed and trust. We’ll get into those essential ingredients some more, but why are those two essential?

ANNE MORRISS: Yeah. Well, the essential pattern that we observed was that the most effective change leaders out there were building trust and speed, and it didn’t seem to be a well-known observation. We all know the phrase, “Move fast and break things,” but the people who were really getting it right were moving fast and fixing things, and that was really our jumping off point. So when we dug into the pattern, what we observed was they were building trust first and then speed. This foundation of trust was what allowed them to fix more things and break fewer.

CURT NICKISCH: Trust sounds like a slow thing, right? If you talk about building trust, that is something that takes interactions, it takes communication, it takes experiences. Does that run counter to the speed idea?

ANNE MORRISS: Yeah. Well, this issue of trust is something we’ve been looking at for over a decade. One of the headlines in our research is it’s actually something we’re building and rebuilding and breaking all the time. And so instead of being this precious, almost farbege egg, it’s this thing that is constantly in motion and this thing that we can really impact when we’re deliberate about our choices and have some self-awareness around where it’s breaking down and how it’s breaking down.

CURT NICKISCH: You said break trust in there, which is intriguing, right? That you may have to break trust to build trust. Can you explain that a little?

ANNE MORRISS:  Yeah, well, I’ll clarify. It’s not that you have to break it in order to build it. It’s just that we all do it some of the time. Most of us are trusted most of the time. Most of your listeners I imagine are trusted most of the time, but all of us have a pattern where we break trust or where we don’t build as much as could be possible.

CURT NICKISCH: I want to talk about speed, this other essential ingredient that’s so intriguing, right? Because you think about solving hard problems as something that just takes a lot of time and thinking and coordination and planning and designing. Explain what you mean by it? And also, just  how we maybe approach problems wrong by taking them on too slowly?

ANNE MORRISS: Well, Curt, no one has ever said to us, “I wish I had taken longer and done less.” We hear the opposite all the time, by the way. So what we really set out to do was to create a playbook that anyone can use to take less time to do more of the things that are going to make your teams and organizations stronger.

And the way we set up the book is okay, it’s really a five step process. Speed is the last step. It’s the payoff for the hard work you’re going to do to figure out your problem, build or rebuild trust, expand the team in thoughtful and strategic ways, and then tell a real and compelling story about the change you’re leading.

Only then do you get to go fast, but that’s an essential part of the process, and we find that either people under emphasize it or speed has gotten a bad name in this world of moving fast and breaking things. And part of our mission for sure was to rehabilitate speed’s reputation because it is an essential part of the change leader’s equation. It can be the difference between good intentions and getting anything done at all.

CURT NICKISCH: You know, the fact that nobody ever tells you, “I wish we had done less and taken more time.” I think we all feel that, right? Sometimes we do something and then realize, “Oh, that wasn’t that hard and why did it take me so long to do it? And I wish I’d done this a long time ago.” Is it ever possible to solve a problem too quickly?

ANNE MORRISS: Absolutely. And we see that all the time too. What we push people to do in those scenarios is really take a look at the underlying issue because in most cases, the solution is not to take your foot off the accelerator per se and slow down. The solution is to get into the underlying problem. So if it’s burnout or a strategic disconnect between what you’re building and the marketplace you’re serving, what we find is the anxiety that people attach to speed or the frustration people attach to speed is often misplaced.

CURT NICKISCH: What is a good timeline to think about solving a problem then? Because if we by default take too long or else jump ahead and we don’t fix it right, what’s a good target time to have in your mind for how long solving a problem should take?

ANNE MORRISS: Yeah. Well, we’re playful in the book and talking about the idea that many problems can be solved in a week. We set the book up five chapters. They’re titled Monday, Tuesday, Wednesday, Thursday, Friday, and we’re definitely having fun with that. And yet, if you count the hours in a week, there are a lot of them. Many of our problems, if you were to spend a focused 40 hours of effort on a problem, you’re going to get pretty far.

But our main message is, listen, of course it’s going to depend on the nature of the problem, and you’re going to take weeks and maybe even some cases months to get to the other side. What we don’t want you to do is take years, which tends to be our default timeline for solving hard problems.

CURT NICKISCH: So you say to start with identifying the problem that’s holding you back, seems kind of obvious. But where do companies go right and wrong with this first step of just identifying the problem that’s holding you back?

ANNE MORRISS: And our goal is that all of these are going to feel obvious in retrospect. The problem is we skip over a lot of these steps and this is why we wanted to underline them. So this one is really rooted in our observation and I think the pattern of our species that we tend to be overconfident in the quality of our thoughts, particularly when it comes to diagnosing problems.

And so we want to invite you to start in a very humble and curious place, which tends not to be our default mode when we’re showing up for work. We convince ourselves that we’re being paid for our judgment. That’s exactly what gets reinforced everywhere. And so we tend to counterintuitively, given what we just talked about, we tend to move too quickly through the diagnostic phase.

CURT NICKISCH: “I know what to do, that’s why you hired me.”

ANNE MORRISS: Exactly. “I know what to do. That’s why you hired me. I’ve seen this before. I have a plan. Follow me.” We get rewarded for the expression of confidence and clarity. And so what we’re inviting people to do here is actually pause and really lean into what are the root causes of the problem you’re seeing? What are some alternative explanations? Let’s get into dialogue with the people who are also impacted by the problem before we start running down the path of solving it.

CURT NICKISCH: So what do you recommend for this step, for getting to the root of the problem? What are questions you should ask? What’s the right thought process? What do you do on Monday of the week?

ANNE MORRISS: In our experience of doing this work, people tend to undervalue the power of conversation, particularly with other people in the organization. So we will often advocate putting together a team of problem solvers, make it a temporary team, really pull in people who have a particular perspective on the problem and create the space, make it as psychologically safe as you can for people to really, as Chris Argyris so beautifully articulated, discuss the undiscussable.

And so the conditions for that are going to look different in every organization depending on the problem, but if you can get a space where smart people who have direct experience of a problem are in a room and talking honestly with each other, you can make an extraordinary amount of progress, certainly in a day.

CURT NICKISCH: Yeah, that gets back to the trust piece.

ANNE MORRISS: Definitely.

CURT NICKISCH: How do you like to start that meeting, or how do you like to talk about it? I’m just curious what somebody on that team might hear in that meeting, just to get the sense that it’s psychologically safe, you can discuss the undiscussable and you’re also focusing on the identification part. What’s key to communicate there?

ANNE MORRISS: Yeah. Well, we sometimes encourage people to do a little bit of data gathering before those conversations. So the power of a quick anonymous survey around whatever problem you’re solving, but also be really thoughtful about the questions you’re going to ask in the moment. So a little bit of preparation can go a long way and a little bit of thoughtfulness about the power dynamic. So who’s going to walk in there with license to speak and who’s going to hold back? So being thoughtful about the agenda, about the questions you’re asking about the room, about the facilitation, and then courage is a very infectious emotion.

So if you can early on create the conditions for people to show up bravely in that conversation, then the chance that you’re going to get good information and that you’re going to walk out of that room with new insight in the problem that you didn’t have when you walked in is extraordinarily high.

CURT NICKISCH: Now, in those discussions, you may have people who have different perspectives on what the problem really is. They also bear different costs of addressing the problem or solving it. You talked about the power dynamic, but there’s also an unfairness dynamic of who’s going to actually have to do the work to take care of it, and I wonder how you create a culture in that meeting where it’s the most productive?

ANNE MORRISS: For sure, the burden of work is not going to be equitably distributed around the room. But I would say, Curt, the dynamic that we see most often is that people are deeply relieved that hard problems are being addressed. So it really can create, and more often than not in our experience, it does create this beautiful flywheel of action, creativity, optimism. Often when problems haven’t been addressed, there is a fair amount of anxiety in the organization, frustration, stagnation. And so credible movement towards action and progress is often the best antidote. So even if the plan isn’t super clear yet, if it’s credible, given who’s in the room and their decision rights and mandate, if there’s real momentum coming out of that to make progress, then that tends to be deeply energizing to people.

CURT NICKISCH: I wonder if there’s an organization that you’ve worked with that you could talk about how this rolled out and how this took shape?

ANNE MORRISS: When we started working with Uber, that was wrestling with some very public issues of culture and trust with a range of stakeholders internally, the organization, also external, that work really started with a campaign of listening and really trying to understand where trust was breaking down from the perspective of these stakeholders?

So whether it was female employees or regulators or riders who had safety concerns getting into the car with a stranger. This work, it starts with an honest internal dialogue, but often the problem has threads that go external. And so bringing that same commitment to curiosity and humility and dialogue to anyone who’s impacted by the problem is the fastest way to surface what’s really going on.

CURT NICKISCH: There’s a step in this process that you lay out and that’s communicating powerfully as a leader. So we’ve heard about listening and trust building, but now you’re talking about powerful communication. How do you do this and why is it maybe this step in the process rather than the first thing you do or the last thing you do?

ANNE MORRISS: So in our process, again, it’s the days of the week. On Monday you figured out the problem. Tuesday you really got into the sandbox in figuring out what a good enough plan is for building trust. Wednesday, step three, you made it better. You created an even better plan, bringing in new perspectives. Thursday, this fourth step is the day we’re saying you got to go get buy-in. You got to bring other people along. And again, this is a step where we see people often underinvest in the power and payoff of really executing it well.

CURT NICKISCH: How does that go wrong?

ANNE MORRISS: Yeah, people don’t know the why. Human behavior and the change in human behavior really depends on a strong why. It’s not just a selfish, “What’s in it for me?” Although that’s helpful, but where are we going? I may be invested in a status quo and I need to understand, okay, if you’re going to ask me to change, if you’re going to invite me into this uncomfortable place of doing things differently, why am I here? Help me understand it and articulate the way forward and language that not only I can understand, but also that’s going to be motivating to me.

CURT NICKISCH: And who on my team was part of this process and all that kind of stuff?

ANNE MORRISS: Oh, yeah. I may have some really important questions that may be in the way of my buy-in and commitment to this plan. So certainly creating a space where those questions can be addressed is essential. But what we found is that there is an architecture of a great change story, and it starts with honoring the past, honoring the starting place. Sometimes we’re so excited about the change and animated about the change that what has happened before or what is even happening in the present tense is low on our list of priorities.

Or we want to label it bad, because that’s the way we’ve thought about the change, but really pausing and honoring what came before you and all the reasonable decisions that led up to it, I think can be really helpful to getting people emotionally where you want them to be willing to be guided by you. Going back to Uber, when Dara Khosrowshahi came in.

CURT NICKISCH: This is the new CEO.

ANNE MORRISS: The new CEO.

CURT NICKISCH: Replaced Travis Kalanick, the founder and first CEO, yeah.

ANNE MORRISS: Yeah, and had his first all-hands meeting. One of his key messages, and this is a quote, was that he was going to retain the edge that had made Uber, “A force of nature.” And in that meeting, the crowd went wild because this is also a company that had been beaten up publicly for months and months and months, and it was a really powerful choice. And his predecessor, Travis was in the room, and he also honored Travis’ incredible work and investment in bringing the company to the place where it was.

And I would use words like grace to also describe those choices, but there’s also an incredible strategic value to naming the starting place for everybody in the room because in most cases, most people in that room played a role in getting to that starting place, and you’re acknowledging that.

CURT NICKISCH: You can call it grace. Somebody else might call it diplomatic or strategic. But yeah, I guess like it or not, it’s helpful to call out and honor the complexity of the way things have been done and also the change that’s happening.

ANNE MORRISS: Yeah, and the value. Sometimes honoring the past is also owning what didn’t work or what wasn’t working for stakeholders or segments of the employee team, and we see that around culture change. Sometimes you’ve got to acknowledge that it was not an equitable environment, but whatever the worker, everyone in that room is bringing that pass with them. So again, making it discussable and using it as the jumping off place is where we advise people to start.

Then you’ve earned the right to talk about the change mandate, which we suggest using clear and compelling language about the why. “This is what happened, this is where we are, this is the good and the bad of it, and here’s the case for change.”

And then the last part, which is to describe a rigorous and optimistic way forward. It’s a simple past, present, future arc, which will be familiar to human beings. We love stories as human beings. It’s among the most powerful currency we have to make sense of the world.

CURT NICKISCH: Yeah. Chronological is a pretty powerful order.

ANNE MORRISS: Right. But again, the change leaders we see really get it right, are investing an incredible amount of time into the storytelling part of their job. Ursula Burns, the Head of Xerox is famous for the months and years she spent on the road just telling the story of Xerox’s change, its pivot into services to everyone who would listen, and that was a huge part of her success.

CURT NICKISCH: So Friday or your fifth step, you end with empowering teams and removing roadblocks. That seems obvious, but it’s critical. Can you dig into that a little bit?

ANNE MORRISS: Yeah. Friday is the fun day. Friday’s the release of energy into the system. Again, you’ve now earned the right to go fast. You have a plan, you’re pretty confident it’s going to work. You’ve told the story of change the organization, and now you get to sprint. So this is about really executing with urgency, and it’s about a lot of the tactics of speed is where we focus in the book. So the tactics of empowerment, making tough strategic trade-offs so that your priorities are clear and clearly communicated, creating mechanisms to fast-track progress. At Etsy, CEO Josh Silverman, he labeled these projects ambulances. It’s an unfortunate metaphor, but it’s super memorable. These are the products that get to speed out in front of the other ones because the stakes are high and the clock is sticking.

CURT NICKISCH: You pull over and let it go by.

ANNE MORRISS: Yeah, exactly. And so we have to agree as an organization on how to do something like that. And so we see lots of great examples both in young organizations and big complex biotech companies with lots of regulatory guardrails have still found ways to do this gracefully.

And I think we end with this idea of conflict debt, which is a term we really love. Leanne Davey, who’s a team scholar and researcher, and anyone in a tech company will recognize the idea of tech debt, which is this weight the organization drags around until they resolve it. Conflict debt is a beautiful metaphor because it is this weight that we drag around and slows us down until we decide to clean it up and fix it. The organizations that are really getting speed right have figured out either formally or informally, how to create an environment where conflict and disagreements can be gracefully resolved.

CURT NICKISCH: Well, let’s talk about this speed more, right? Because I think this is one of those places that maybe people go wrong or take too long, and then you lose the awareness of the problem, you lose that urgency. And then that also just makes it less effective, right? It’s not just about getting the problem solved as quickly as possible. It’s also just speed in some ways helps solve the problem.

ANNE MORRISS: Oh, yeah. It really is the difference between imagining the change you want to lead and really being able to bring it to life. Speed is the thing that unlocks your ability to lead change. It needs a foundation, and that’s what Monday through Thursday is all about, steps one through four, but the finish line is executing with urgency, and it’s that urgency that releases the system’s energy, that communicates your priorities, that creates the conditions for your team to make progress.

CURT NICKISCH: Moving fast is something that entrepreneurs and tech companies certainly understand, but there’s also this awareness that with big companies, the bigger the organization, the harder it is to turn the aircraft carrier around, right? Is speed relative when you get at those levels, or do you think this is something that any company should be able to apply equally?

ANNE MORRISS: We think this applies to any company. The culture really lives at the level of team. So we believe you can make a tremendous amount of progress even within your circle of control as a team leader. I want to bring some humility to this and careful of words like universal, but we do think there’s some universal truths here around the value of speed, and then some of the byproducts like keeping fantastic people. Your best people want to solve problems, they want to execute, they want to make progress and speed, and the ability to do that is going to be a variable in their own equation of whether they stay or they go somewhere else where they can have an impact.

CURT NICKISCH: Right. They want to accomplish something before they go or before they retire or finish something out. And if you’re able to just bring more things on the horizon and have it not feel like it’s going to be another two years to do something meaningful.

ANNE MORRISS: People – I mean, they want to make stuff happen and they want to be around the energy and the vitality of making things happen, which again, is also a super infectious phenomenon. One of the most important jobs of a leader, we believe, is to set the metabolic pace of their teams and organizations. And so what we really dig into on Friday is, well, what does that look like to speed something up? What are the tactics of that?

CURT NICKISCH: I wonder if that universal truth, that a body in motion stays in motion applies to organizations, right? If an organization in motion stays in motion, there is something to that.

ANNE MORRISS: Absolutely.

CURT NICKISCH: Do you have a favorite client story to share, just where you saw speed just become a bit of a flywheel or just a positive reinforcement loop for more positive change at the organization?

ANNE MORRISS: Yeah. We work with a fair number of organizations that are on fire. We do a fair amount of firefighting, but we also less dramatically do a lot of fire prevention. So we’re brought into organizations that are working well and want to get better, looking out on the horizon. That work is super gratifying, and there is always a component of, well, how do we speed this up?

What I love about that work is there’s often already a high foundation of trust, and so it’s, well, how do we maintain that foundation but move this flywheel, as you said, even faster? And it’s really energizing because often there’s a lot of pent-up energy that… There’s a lot of loyalty to the organization, but often it’s also frustration and pent-up energy. And so when that gets released, when good people get the opportunity to sprint for the first time in a little while, it’s incredibly energizing, not just for us, but for the whole organization.

CURT NICKISCH: Anne, this is great. I think finding a way to solve problems better but also faster is going to be really helpful. So thanks for coming on the show to talk about it.

ANNE MORRISS:  Oh, Curt, it was such a pleasure. This is my favorite conversation. I’m delighted to have it anytime.

HANNAH BATES: That was entrepreneur, leadership coach, and author Anne Morriss – in conversation with Curt Nickisch on HBR IdeaCast.

We’ll be back next Wednesday with another hand-picked conversation about business strategy from Harvard Business Review. If you found this episode helpful, share it with your friends and colleagues, and follow our show on Apple Podcasts, Spotify, or wherever you get your podcasts. While you’re there, be sure to leave us a review.

When you’re ready for more podcasts, articles, case studies, books, and videos with the world’s top business and management experts, you’ll find it all at HBR.org.

This episode was produced by Mary Dooe, Anne Saini, and me, Hannah Bates. Ian Fox is our editor. Special thanks to Rob Eckhardt, Maureen Hoch, Erica Truxler, Ramsey Khabbaz, Nicole Smith, Anne Bartholomew, and you – our listener. See you next week.

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1.5: Teaching Mathematics Through Problem Solving

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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 ”.

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7.4: Solving problems with work and energy

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• Page ID 103477

• Peter G. Steeneken
• Delft University of Technology

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In this chapter two methods to solve dynamics problems using the concepts of work and energy have been described: the principle of work and energy and the concept of energy conservation. Both methods provide one scalar equation, Equation 7.11 and Equation 7.34, that can be used to determine one unknown scalar variable. For example, this unknown scalar variable can be:

• The magnitude of a final or initial velocity
• A final or initial position coordinate
• The work of a force, which can sometimes also be used to determine the magnitude of the force

The advantage of energy methods is that they allow solving a dynamics problems, without requiring one to solve differential equations. Moreover, under certain conditions (energy conservation) they provide solutions even if the path between the initial and final state is unknown, making the solutions also more generally valid.

Solving the problem then proceeds along the following steps:

• Sketch the point masses, massless mechanisms, force vectors and constraints.
• Choose and draw a suitable coordinate system (CS).
• Determine the constraint equations.
• Determine the kinetic energy function of all point masses in terms of the velocities.
• Determine the work or potential energy function for all forces.
• Determine the change in kinetic energy, change in potential energy, and/or the work done on masses by forces between the initial and final condition.
• Write down the equation for the principle of work and energy, or the conservation of energy equation.
• Determine the unknown scalar variable by solving the energy equation.

Energy methods only give the scalar values of the positions and velocity and not their directions. For instance, a mass with velocity \(v\) has the same kinetic energy as a mass with velocity \(-v\), such that the direction of the velocity cannot be determined from the kinetic energy. Similarly, if the potential energy of a spring is calculated by energy methods, it cannot be determined if it is compressed or extended. Careful evaluation of the problem is needed, based on considerations other than energy, to determine those directions and to select the right solution.