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The art of problem solving 7th edition

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The Art of Problem Solving A Resource for the Mathematics Teacher

• Alfred S. Posamentier - Mercy College, New York, USA
• Wolfgang Schulz
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The Art of Effective Problem Solving: A Step-by-Step Guide

Author: Daniel Croft

Daniel Croft is an experienced continuous improvement manager with a Lean Six Sigma Black Belt and a Bachelor's degree in Business Management. With more than ten years of experience applying his skills across various industries, Daniel specializes in optimizing processes and improving efficiency. His approach combines practical experience with a deep understanding of business fundamentals to drive meaningful change.

Whether we realise it or not, problem solving skills are an important part of our daily lives. From resolving a minor annoyance at home to tackling complex business challenges at work, our ability to solve problems has a significant impact on our success and happiness. However, not everyone is naturally gifted at problem-solving, and even those who are can always improve their skills. In this blog post, we will go over the art of effective problem-solving step by step.

You will learn how to define a problem, gather information, assess alternatives, and implement a solution, all while honing your critical thinking and creative problem-solving skills. Whether you’re a seasoned problem solver or just getting started, this guide will arm you with the knowledge and tools you need to face any challenge with confidence. So let’s get started!

Problem Solving Methodologies

Individuals and organisations can use a variety of problem-solving methodologies to address complex challenges. 8D and A3 problem solving techniques are two popular methodologies in the Lean Six Sigma framework.

Methodology of 8D (Eight Discipline) Problem Solving:

The 8D problem solving methodology is a systematic, team-based approach to problem solving. It is a method that guides a team through eight distinct steps to solve a problem in a systematic and comprehensive manner.

The 8D process consists of the following steps:

• Form a team: Assemble a group of people who have the necessary expertise to work on the problem.
• Define the issue: Clearly identify and define the problem, including the root cause and the customer impact.
• Create a temporary containment plan: Put in place a plan to lessen the impact of the problem until a permanent solution can be found.
• Identify the root cause: To identify the underlying causes of the problem, use root cause analysis techniques such as Fishbone diagrams and Pareto charts.
• Create and test long-term corrective actions: Create and test a long-term solution to eliminate the root cause of the problem.
• Implement and validate the permanent solution: Implement and validate the permanent solution’s effectiveness.
• Prevent recurrence: Put in place measures to keep the problem from recurring.
• Recognize and reward the team: Recognize and reward the team for its efforts.

Download the 8D Problem Solving Template

A3 Problem Solving Method:

The A3 problem solving technique is a visual, team-based problem-solving approach that is frequently used in Lean Six Sigma projects. The A3 report is a one-page document that clearly and concisely outlines the problem, root cause analysis, and proposed solution.

The A3 problem-solving procedure consists of the following steps:

• Determine the issue: Define the issue clearly, including its impact on the customer.
• Perform root cause analysis: Identify the underlying causes of the problem using root cause analysis techniques.
• Create and implement a solution: Create and implement a solution that addresses the problem’s root cause.
• Monitor and improve the solution: Keep an eye on the solution’s effectiveness and make any necessary changes.

Subsequently, in the Lean Six Sigma framework, the 8D and A3 problem solving methodologies are two popular approaches to problem solving. Both methodologies provide a structured, team-based problem-solving approach that guides individuals through a comprehensive and systematic process of identifying, analysing, and resolving problems in an effective and efficient manner.

Step 1 – Define the Problem

The definition of the problem is the first step in effective problem solving. This may appear to be a simple task, but it is actually quite difficult. This is because problems are frequently complex and multi-layered, making it easy to confuse symptoms with the underlying cause. To avoid this pitfall, it is critical to thoroughly understand the problem.

To begin, ask yourself some clarifying questions:

• What exactly is the issue?
• What are the problem’s symptoms or consequences?
• Who or what is impacted by the issue?
• When and where does the issue arise?

Answering these questions will assist you in determining the scope of the problem. However, simply describing the problem is not always sufficient; you must also identify the root cause. The root cause is the underlying cause of the problem and is usually the key to resolving it permanently.

Try asking “why” questions to find the root cause:

• What causes the problem?
• Why does it continue?
• Why does it have the effects that it does?

By repeatedly asking “ why ,” you’ll eventually get to the bottom of the problem. This is an important step in the problem-solving process because it ensures that you’re dealing with the root cause rather than just the symptoms.

Once you have a firm grasp on the issue, it is time to divide it into smaller, more manageable chunks. This makes tackling the problem easier and reduces the risk of becoming overwhelmed. For example, if you’re attempting to solve a complex business problem, you might divide it into smaller components like market research, product development, and sales strategies.

To summarise step 1, defining the problem is an important first step in effective problem-solving. You will be able to identify the root cause and break it down into manageable parts if you take the time to thoroughly understand the problem. This will prepare you for the next step in the problem-solving process, which is gathering information and brainstorming ideas.

Step 2 – Gather Information and Brainstorm Ideas

Gathering information and brainstorming ideas is the next step in effective problem solving. This entails researching the problem and relevant information, collaborating with others, and coming up with a variety of potential solutions. This increases your chances of finding the best solution to the problem.

Begin by researching the problem and relevant information. This could include reading articles, conducting surveys, or consulting with experts. The goal is to collect as much information as possible in order to better understand the problem and possible solutions.

Next, work with others to gather a variety of perspectives. Brainstorming with others can be an excellent way to come up with new and creative ideas. Encourage everyone to share their thoughts and ideas when working in a group, and make an effort to actively listen to what others have to say. Be open to new and unconventional ideas and resist the urge to dismiss them too quickly.

Finally, use brainstorming to generate a wide range of potential solutions. This is the place where you can let your imagination run wild. At this stage, don’t worry about the feasibility or practicality of the solutions; instead, focus on generating as many ideas as possible. Write down everything that comes to mind, no matter how ridiculous or unusual it may appear. This can be done individually or in groups.

Once you’ve compiled a list of potential solutions, it’s time to assess them and select the best one. This is the next step in the problem-solving process, which we’ll go over in greater detail in the following section.

Step 3 – Evaluate Options and Choose the Best Solution

Once you’ve compiled a list of potential solutions, it’s time to assess them and select the best one. This is the third step in effective problem solving, and it entails weighing the advantages and disadvantages of each solution, considering their feasibility and practicability, and selecting the solution that is most likely to solve the problem effectively.

To begin, weigh the advantages and disadvantages of each solution. This will assist you in determining the potential outcomes of each solution and deciding which is the best option. For example, a quick and easy solution may not be the most effective in the long run, whereas a more complex and time-consuming solution may be more effective in solving the problem in the long run.

Consider each solution’s feasibility and practicability. Consider the following:

• Can the solution be implemented within the available resources, time, and budget?
• What are the possible barriers to implementing the solution?
• Is the solution feasible in today’s political, economic, and social environment?

You’ll be able to tell which solutions are likely to succeed and which aren’t by assessing their feasibility and practicability.

Finally, choose the solution that is most likely to effectively solve the problem. This solution should be based on the criteria you’ve established, such as the advantages and disadvantages of each solution, their feasibility and practicability, and your overall goals.

It is critical to remember that there is no one-size-fits-all solution to problems. What is effective for one person or situation may not be effective for another. This is why it is critical to consider a wide range of solutions and evaluate each one based on its ability to effectively solve the problem.

Step 4 – Implement and Monitor the Solution

When you’ve decided on the best solution, it’s time to put it into action. The fourth and final step in effective problem solving is to put the solution into action, monitor its progress, and make any necessary adjustments.

To begin, implement the solution. This may entail delegating tasks, developing a strategy, and allocating resources. Ascertain that everyone involved understands their role and responsibilities in the solution’s implementation.

Next, keep an eye on the solution’s progress. This may entail scheduling regular check-ins, tracking metrics, and soliciting feedback from others. You will be able to identify any potential roadblocks and make any necessary adjustments in a timely manner if you monitor the progress of the solution.

Finally, make any necessary modifications to the solution. This could entail changing the solution, altering the plan of action, or delegating different tasks. Be willing to make changes if they will improve the solution or help it solve the problem more effectively.

It’s important to remember that problem solving is an iterative process, and there may be times when you need to start from scratch. This is especially true if the initial solution does not effectively solve the problem. In these situations, it’s critical to be adaptable and flexible and to keep trying new solutions until you find the one that works best.

To summarise, effective problem solving is a critical skill that can assist individuals and organisations in overcoming challenges and achieving their objectives. Effective problem solving consists of four key steps: defining the problem, generating potential solutions, evaluating alternatives and selecting the best solution, and implementing the solution.

You can increase your chances of success in problem solving by following these steps and considering factors such as the pros and cons of each solution, their feasibility and practicability, and making any necessary adjustments. Furthermore, keep in mind that problem solving is an iterative process, and there may be times when you need to go back to the beginning and restart. Maintain your adaptability and try new solutions until you find the one that works best for you.

• Novick, L.R. and Bassok, M., 2005.  Problem Solving . Cambridge University Press.

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Daniel Croft

Hi im Daniel continuous improvement manager with a Black Belt in Lean Six Sigma and over 10 years of real-world experience across a range sectors, I have a passion for optimizing processes and creating a culture of efficiency. I wanted to create Learn Lean Siigma to be a platform dedicated to Lean Six Sigma and process improvement insights and provide all the guides, tools, techniques and templates I looked for in one place as someone new to the world of Lean Six Sigma and Continuous improvement.

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About the Art of Problem Solving Initiative

Founded in 2004, the Art of Problem Solving Initiative, Inc. was created by people who love math and love teaching to help students access the study of advanced mathematics.

The Initiative began by running the USA Mathematical Talent Search , a nationwide math contest sponsored by the National Security Agency which continues to run to this day. In each round, this contest gives students a full month to work on five challenging proof-based problems. Students then get individual feedback on their work, including comments on their mathematical reasoning and on their proof-writing skills. By taking away the usual time pressures seen in many math contests, students have the opportunity to go more deeply and to explore.

For many years, the Initiative also ran the Local Programs Initiative which provided fiscal sponsorship to math circles and clubs across the country.

Now, the main project of the Initiative is Bridge to Enter Advanced Mathematics (BEAM) , a project to help underserved students find a realistic pathway towards becoming scientists, mathematicians, engineers, and programmers. During the summer after 7th grade, students are invited to participate in a free three-week residential program on a college campus where they gain the academic and social/emotional preparation to succeed in future programs for advanced study. They then receive academic advising throughout 8th grade and high school to help them and their families find the best opportunities for their educations.

The Art of Problem Solving Initiative receives support from Art of Problem Solving (AoPS) , which develops resources for high-performing middle and high school students including the largest online community of avid math students in the English-speaking world. The AoPS online school has trained many winners of major national mathematics competitions, including several gold medalists at the International Math Olympiad, Davidson Fellows, and winners of the Intel and Siemens Talent Search competitions.

The Art of Problem Solving Initiative is a 501(c)(3) charitable organization. It continues to be led by expert mathematicians and educators to this day. Please consider joining our contributors by donating to support advanced math education.

Financial Information

The Art of Problem Solving Initiative, Inc. makes its financial statements available for public inspection in the interest of greater transparency. Please see below to learn more about the organization.

• Tax return (Form 990)
• Audited financial statements
• Tax return (Form 990) (Note that due to a change in fiscal year these cover a shortened period.)

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Review by Pamela Brittain , Common Sense Education | Updated August 2019

Art of Problem Solving

Variety of math content supports struggling and advanced learners

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Community rating, privacy rating, subjects & topics, take a look inside 8 images.

Pros : Presents information in multiple ways; courses are organized specifically by grade level.

Cons : The forums are a bit hard to navigate; continued use of the program could get a bit pricy over time.

Bottom Line : Separate sections for younger and older students help to teach kids at all levels, from foundational skills up to Math Olympiad training.

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The Art of Problem Solving allows students to focus on math concepts that they're struggling with or (for advanced students) that offer a challenge. The site presents the materials as self-paced for younger students or as online, interactive courses for older students. There's also a large collection of additional resources available on the site, including videos and an extensive forum (over eight million posts and counting) where students discuss all sorts of math concepts. 

The site is best designed for students wishing to work independently on their math knowledge, although some of the materials presented -- such as the videos -- may be good for a teacher to use with a whole class to introduce a topic or give a different perspective on it. If you're a Math Olympiad coach or have students preparing for math contests, you'll find the videos of sample test questions very helpful; many of the resources found here are especially geared toward kids who are preparing to compete at a national level.

The Art of Problem Solving has added a lot of math content over the past few years and now includes videos, interactive courses, and a fun, game-based academy for younger students. The site is divided into different sections, each focusing on a different area of math and different grade levels. The main divisions are AoPS Online, Beast Academy, and AoPS Academy.

AoPS Online provides older students with lessons on everything from pre-algebra foundational skills up to Math Olympiad training. There's also a substantial forum where students can interact with one another to ask -- and answer -- various math problems. The Beast Academy is for younger students and has a fun interface with comic characters, videos, games, and more for grades 2 to 5. Finally, the AoPS Academy is an in-person academy that students can attend. 

Overall, the Art of Problem Solving provides a variety of resources to help struggling students succeed and to encourage and build enrichment for students to challenge themselves. The fact that it's all presented together in one place may also have the bonus effect of encouraging once struggling students to take on greater and greater challenges -- and perhaps be the next Math Olympians. 

The choice of videos, books, forums, online interactive classes, and games (for the younger students) means that students have a number of different ways to learn the material. The Beast Academy, in particular, is very good at providing students with different learning options. The bright colors and fun characters also lend an air of friendless to the site without losing the focus on teaching. 

The forum promotes collaboration among students, but it's hard to navigate, and some of the posts are less about math and more about "kids being kids" in an online environment. But for those who take the time to explore, there's a rich source of information to be found.

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A variety of content formats gives students of different abilities multiple options for accessing the materials. The forum, though hard to navigate, encourages kids to engage with one another in a safe, friendly way. 

Math content covers a large number of concepts and is structured differently by grade level: webinar lectures for older students and games and comic book characters for young kids.

The Beast Academy supplies kids with lots of hints, and online, interactive classes give students direct access to the teacher. Digging through the forum may provide additional help.

Strong program with emphasis on deep understanding of concepts and love of math. Would highly recommend above most curriculums I have tried.

The K-5 program is strong and could easily be used in any k-6 program. The content is challenging but so strongly done that any level of student could use the program with support.

The art of problem solving program for 6-12 is very challenging. It could be used for all students, but again, would need support. The online classes are excellent, books alone would need strong classroom support. Clearly aimed at high functioning students. This is a math program that goes beyond basic understanding and rote memorization to emphasize true, deep understanding and complex thinking. There are a plethora of problems available and a rich online community.

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Art of Problem Solving (AoPS) Prealgebra F Fourth Printing New Edition

• ISBN-10 1934124214
• ISBN-13 978-1934124215
• Edition F Fourth Printing New
• Publisher AoPS Incorporated
• Publication date August 9, 2011
• Language English
• Dimensions 10.38 x 8.38 x 0.5 inches
• Print length 608 pages
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• Publisher ‏ : ‎ AoPS Incorporated; F Fourth Printing New edition (August 9, 2011)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 608 pages
• ISBN-10 ‏ : ‎ 1934124214
• ISBN-13 ‏ : ‎ 978-1934124215
• Item Weight ‏ : ‎ 2.95 pounds
• Dimensions ‏ : ‎ 10.38 x 8.38 x 0.5 inches
• #207 in Textbooks (Special Features Stores)

About the authors

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.

David Patrick

David Patrick has been an author and curriculum developer at Art of Problem Solving since 2004. He is the author of Art of Problem Solving's Introduction to Counting & Probability, Intermediate Counting & Probability, and Calculus textbooks, and co-author of AoPS's new Prealgebra textbook. He is also an instructor in AoPS's widely-acclaimed online school for high performing middle and high school students.

Dave earned the sole perfect score on the American High School Mathematics Examination (AHSME) in 1988 and was a USA Mathematical Olympiad winner that year. He attended the Research Science Institute (RSI) in 1987, and the Math Olympiad Summer Program in 1988, where he first met fellow student Richard Rusczyk. He also finished in the top 10 on the Putnam exam in 1991. Dave graduated from Carnegie Mellon in 1992 with a BS in Mathematics/Computer Science and an MS in Mathematics. He went on to earn his Ph.D. in mathematics from MIT in 1997. He was an acting Assistant Professor at the University of Washington from 1997 to 2001. Dave is originally from Western New York and is an alumnus of the SUNY Buffalo Gifted Math Program.

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Customers find the explanations in the book excellent and easy to understand. They also say it's a fantastic approach to learning. However, opinions are mixed on the look, with some finding it great and not fancy, while others say it doesn't look good and the layout is bland.

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Customers find the explanations in the book excellent. They say it's a fantastic approach to learning, and the concepts are very easy to understand. Readers also mention the book covers seemingly simple topics with depth and insight. They appreciate the plenty of review and challenge problems that keep them coming back.

"...videos, many aligned to specific topics in the textbooks, and there's a pretty cool (currently free) tool called 'Alcumus' that challenges you with..." Read more

"...is always eager to learn something new and this really satisfies his need to be challenged ...." Read more

"...It teaches different tips and tricks to manipulate numbers in your head easier. (My son appreciates this since he hates writing, LOL)..." Read more

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