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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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• Art of Problem Solving

Art of Problem Solving ( AoPS ) is an educational resources company founded by Richard Rusczyk in 2003.

• 1.1 Mission
• 1.2 Tradition of Excellence
• 1.3 The Staff
• 2.1 AoPS Wiki
• 2.2.1 Subject textbooks
• 2.2.2 Competition preparation books
• 2.2.3 Other math books
• 2.3 AoPS Online School
• 2.4 AoPS Forums
• 2.5 AoPS: For The Win
• 2.6 TeX/LaTeX Resources
• 2.7 Math contest problem database
• 2.8 Articles
• 3 Philanthropy

The Company

The main goal of AoPS is to create interactive educational opportunities for avid students of mathematics . As time goes on, AoPS is reaching out to students of other problem solving disciplines as well, including informatics , physics , programming, and others.

Tradition of Excellence

The accomplishments of every student are unique, and there is no way to measure that success. However, we try to record and celebrate achievements of AoPS students, faculty, and community members .

For a list of all the current staff at AoPS and some previous staff go to the AoPS Administrators page.

Resources on AoPS

You're in the AoPS Wiki now!

• AoPSWiki:Table of Contents -- a basic guide to AoPSWiki content
• Academic competitions including a huge List of mathematics competitions
• Academic scholarships including a large list of Mathematics scholarships

Subject textbooks

AoPS subject texts provide instruction to excellent students of mathematics. This modern curriculum is both comprehensive and challenging enough for brilliant young mathematical minds.

• Introduction to Algebra
• Introduction to Counting & Probability
• Introduction to Geometry
• Introduction to Number Theory
• Intermediate Algebra
• Intermediate Counting & Probability
• Precalculus

Competition preparation books

The Art of Problem Solving competition preparation books cover a variety of topics of interest to students of mathematics interested in competitive math.

• Art of Problem Solving Volume 1: the Basics
• Art of Problem Solving Volume 2: and Beyond
• Competition Math for Middle School

Other math books

AoPS sells numerous other math books at the AoPS Bookstore . Many of these books focus on competitions such as the AMC , the Mandelbrot Competition , Mathcounts , or MOEMS .

AoPS Online School

The AoPS Online School hosts math classes primarily for bright middle and high school students. Students of the online math school include winners of nearly every major mathematics competition in the U.S.

AoPS Forums

The AoPS Forums are a place where students and problem solvers of all ages can discuss mathematics and problem solving. Visit AoPSWiki:AoPS forums for more details.

AoPS: For The Win

For The Win is an online interactive game for students to participate in activities similar to the MATHCOUNTS Countdown Round.

TeX/LaTeX Resources

The AoPS Forums are LaTeX -enabled, allowing users to post nice mathematics.

Math contest problem database

AoPS members help organize a large and well-organized [aops.com/resources.php list of problems and solutions] from mathematics competitions all over the world.

AoPS resources include numerous very good articles for students, parents, and teachers.

Philanthropy

Many of the features of the Art of Problem Solving website are made available for free so that every student interested in mathematics can use them.

Additionally, AoPS instructors provide hundreds of hours of teaching and support time for the San Diego Math Circle .

The Art of Problem Solving Foundation is a nonprofit corporation founded by AoPS CEO Richard Rusczyk with largely the same fundamental mission as AoPS itself, but with greater focus on providing educational resources where they are lacking. AoPS donates a portion of its book and class sales to the foundation. A link can be found here . Make sure it is .org if you want to see the foundation website, otherwise, you'll see the homepage of the current website you're on.

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The Art of Problem Solving Math

The Art of Problem Solving (AoPS) math courses for grades five through twelve were designed for high-performing math students. The publisher says on their website ,

 We present a much broader and deeper exploration of challenging mathematics than a typical math curriculum and show students how to apply their knowledge and problem-solving skills to difficult problems. We help students learn the critical problem-solving skills necessary for success at mathematics competitions (such as MATHCOUNTS and the AMC), top universities, and competitive careers.

Their courses cover much more than typical math courses for middle through high school. They have courses that cover the standard sequence at advanced levels, plus other courses that take students deeper into the math needed for physics, engineering, computer science, and other math-based careers.

 AoPS students often work a few years ahead of other students, which means that capable fifth or sixth graders might start with Prealgebra . Note that AoPS is the publisher of Beast Academy math courses for grades one through five, and those courses prepare students to move right into AoPS Prealgebra.

Courses and Format Options

AoPS lists five courses as part of their “Introductory Curriculum” for students up through tenth grade : Prealgebra, Introduction to Algebra, Introduction to Counting & Probability, Introduction to Geometry , and Introduction to Number Theory . Their “Intermediate Curriculum” for advanced high school students includes Intermediate Algebra, Intermediate Counting & Probability, Precalculus , and Calculus . Even so, students in a traditional program can still use these courses following a more typical timeline and concentrating on the required courses.

The website page for each course has two free diagnostic tests (PDFs) that help determine whether a student has the prerequisites for the course or whether they already have mastered what the course covers. These tests are accessed by clicking on “Are You Ready?” and “Do You Need This?” on each course’s description page.

Students do not need to complete all books in the series, but if they start the series in sixth grade, they should be able to complete most of them. Students who want to participate in math competitions might also be interested in AoPS books written specifically for that purpose: Competition Math for Middle School; the Art of Problem Solving, Volume 1: the Basics ; and the Art of Problem Solving, Volume 2: and Beyond . (The titles of the last two books do not begin with capital T.)

AoPS sells printed or online books or a combination of both. They also offer live, online options for all courses and a self-paced-online option for Prealgebra and Introductory Algebra A. (The online courses might be a great way for eager students to find the community support they need to enter competitions.)

The printed textbooks have separate solutions manuals with worked-out solutions for every problem. The online books include the solutions, and they also integrate the textbooks with interactions with the AoPS community, Alcumus (described below), and the free videos (also described below). The textbooks vary in length; those for the standard courses (except Calculus ) run from 528 to 720 pages, while other courses have from 256 to 400 pages.

Free videos are available online for Prealgebra, Introduction to Algebra , and Introduction to Counting & Probability . You can view these without having to pay or register. The videos do not replace the textbooks or online classes but supplement them. There are one or more videos for each lesson, all taught by Richard Rusczyk, a very engaging presenter as well as a former USA Mathematical Olympiad winner. I highly recommend watching them.

How the Courses Work

The courses divide the content into chapters, with several lessons within each chapter. Each lesson begins either with brief instruction or a set of three or more problems. Students should try to solve the problems on their own. The lesson continues with thorough explanations for how to solve each problem, and this is where most of the instruction is presented. This strategy very much reflects the title of the series, the Art of Problem Solving—students are focused on developing problem-solving skills as well as accuracy.

After studying the solutions, students have another set of problems to solve, a few of which are drawn from advanced math exams (no longer in use), such as the AHSME (American High School Mathematics Examination).

Lessons often use blue boxes to highlight key concepts, important ideas, and warnings about common mistakes.

There are Review Problems at the end of each chapter but no quizzes or tests for any of these courses. (The second diagnostic test for each course, titled “Do You Need This?,” could function as a final exam if needed.) The publisher’s explanation to me regarding this was: “Since our curriculum focuses on teaching students mathematical concepts and problem-solving skills, we believe that students who are using our textbooks have mastered the material if they can successfully solve the Review Problems at the end of each chapter.”

Alcumus, AoPS’s online learning system, is available for free to all students, even if they have not purchased any AoPS course. Alcumus adapts to the student’s performance, giving them problems to solve that are appropriate for their level—problems to solve in addition to those in their course. Alcumus can be used alongside the Introductory Curriculum courses, whether in print or online. (Students using courses from other publishers for pre-algebra through geometry should also find the program useful.) Alcumus provides ambitious students with work that will both reinforce and stretch their skills.

The AoPS website offers many other resources for advanced math students, including information about competitions, online forums, and training for competitions.

AoPS math courses should be fantastic for avid math students who are eager to learn and go deeper, but they also offer excellent and thorough instruction for the average student.

Pricing Information

When prices appear, please keep in mind that they are subject to change. Click on links where available to verify price accuracy.

See the publisher's website for options and prices.

Core Curricula

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• Bible & Religion
• Catholic Curricula
• Composition & Grammar
• Early Learning / Preschool
• Foreign Language
• Handwriting
• History & Geography
• Grade Level Packages & Courses
• Math Supplements
• Phonics & Reading
• Spelling & Vocabulary
• Unit Studies & All-In-One Programs
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Instant Key

• Need For Parent or Teacher Instruction: low
• Learning Environment: all situations
• Grade Level: grades 5-12
• Special Audience: gifted
• Educational Methods: traditional activity pages or exercises, multisensory, highly structured, critical thinking
• Technology: video, supplemental digital content, other ebook, online
• Educational Approaches: eclectic
• Religious Perspective: secular

Publisher's Info

• Art of Problem Solving, Inc.
• PO Box 2185
• Alpine, CA 91903
• [email protected]
• https://artofproblemsolving.com/

Note: Publishers, authors, and service providers never pay to be reviewed. They do provide free review copies or online access to programs for review purposes.

Disclosure of Material Connection: Some of the links in the post above are "affiliate links." This means if you click on the link and purchase the item, I will receive an affiliate commission. Regardless, I only recommend products or services that I believe will add value to my readers. I am disclosing this in accordance with the Federal Trade Commission's 16 CFR, Part 255 "Guidelines Concerning the Use of Endorsements and Testimonials in Advertising."

Art of Problem Solving Math Books Review

What is art of problem solving.

Founded in 1993 by former USA Math Olympiad winner Richard Rusczyk, Art of Problem Solving (AoPS) is a company that produces rigorous math instruction courses and products that can help outstanding math students develop a more thorough understanding of math concepts, as well as help prepare them for success in math competitions. 

From textbooks to online classes to physical learning centers, AoPS offers a variety of educational products and services that can help challenge kids, deepening their knowledge and strengthening their mathematical thinking.

AoPS Math textbooks

Art of Problem Solving has created a series of textbooks for middle and high school math textbooks that are designed to give outstanding math students a deeper and more rigorous curriculum in math. 

Originally designed to help talented math students prepare for competitions, over the years AoPS’s textbook line has expanded to offer full curriculums in middle and high school math courses, and their problem-based and rigorous approach to math has made them very popular with parents across the world as a top enrichment option. 

What Grades and Math Subjects does AoPS Math cover?

Art of Problem Solving textbooks cover middle and high school math, as well as competition prep.  

Generally speaking, the AoPS math textbooks can be broken down into two curricula- introductory and advanced – that roughly correspond to most middle and high school math programs (in terms of overall scope, that is). 

Parents of younger math enthusiasts should note that Art of Problem Solving covers elementary school math (Grades 1-6) in their Beast Academy series, which you can read about in our review .

Introductory Curriculum (Middle School)

Advanced Curriculum (High School) 

When taken as a whole, Art of Problem Solving’s math textbooks cover the topics included in most US Math curricula, as well as touching on a few topics that aren’t usually covered in most public high school programs. 

That said, the point isn’t really to get kids learning college level math or a curriculum beyond high school math, but instead to get students to develop their problem solving skills and develop more creative and flexible mathematical thinking, to get them to recognize and appreciate different approaches to problem solving, as well as getting a better understanding of the why of math, rather than just focusing on how to compute problems. 

As such, AoPS’s curricula tend to go deeper into your typical middle and high school math topics, letting kids examine concepts more rigorously, more thoroughly and with more challenging problems than they would otherwise be able to do in other math courses. 

Art of Problem Solving Contest Prep

In addition to their more academically-focused textbooks, Art of Problem Solving also offers a variety of books designed to further enrich exceptional students or help with preparing for math contests and Olympiads.

These books generally tend to work on developing stronger problem solving skills, going far deeper into various concepts and exploring far more challenging questions and problems, while introducing various approaches for understanding and solving them quickly and effectively.

Geared more for gifted enrichment and contests preparation, each of these books tend to go over a greater variety of concepts and topics, touching on concepts in Geometry, Algebra, Number theory and more, and aren’t really bound to any linear curriculum. 

In addition, the problem sets, geared as they are to helping students prepare for national tournaments and contests, are far more challenging and in-depth than would be expected of even an advanced middle or high school course.

For these reasons we don’t usually think this series is where parents should necessarily start off when working on math at home, but in our experience we do feel they are great supplements to the main textbooks and can be excellent for enrichment purposes and preparing for contests.

How Art of Problem Solving Teaches Math

Aops pedagogical approach.

Art of Problem solving is a big believer in teaching through solving problems. 

The books consequently include a wide variety of problems, many of which kids will have never encountered before.

In fact, some come directly from various math competitions such as:

• The American Mathematics Competitions (AMC)  
• The Harvard-MIT Math Tournament

The general idea is that by getting kids to work through problems themselves, and more importantly discovering how to solve certain problems, kids will develop a deeper understanding of the material. 

As a result, AoPS Math textbooks are quite problem set heavy.

Explanations of each concept are quite short and to the point and are followed by a good deal of exercises for students to try out on their own.

When introducing these textbooks, parents should expect that kids will have to think things through a bit more and work out the answers themselves without a lot of hand holding or spoon feeding, and that there will be a heavier emphasis on logic and proof than other curricula. 

All this really drives home Art of Problem Solving’s place as a resource for outstanding or talented math students who don’t need a lot of time or explanation to grasp the material. 

Consequently, students who are less adept at math may find the instructions a little too short and too quick and may need extra help in order prevent getting frustrated by skill and knowledge gaps as the exercises come rolling in

Lesson structure

Regardless of the book in question, Age of Learning’s lessons tend to follow a particular format.

The books are made up of several chapters, each of which covers a particular topic within the subject and contains several sections. 

Each section is then typically broken down into various related concepts, an overview of the types of problems kids may come across (both common and uncommon) and often the various factors that can affect outcomes.

In Introduction to Algebra, for example, when discussing multivariable linear equations, the chapter is divided up into an introduction, a discussion of substitution, elimination, some word problems, common and uncommon problem sets, different variables and so on.

As kids go through their lessons, they are given lots of examples to try and lessons tend to work through some of them step-by step in a fairly in-depth and rigorous manner to demonstrate concepts. 

Sections typically end with a variety of exercises for that section and, at the end of each chapter, there are review and challenge problems. 

Review problems go over and test what the student has learned with similar problems, while challenge problems go a step further and test mastery of the material with far more challenging questions. 

If kids get stuck, there are always hints and solutions that are helpfully included in the back of the book ( no cheating !)

Look and feel 

As you might expect from a problem solving and word problem-heavy methodology, these textbooks contain lots of typical math diagrams and pictures floating about to go along with and illustrate the word problems.

AoPS textbooks also tend to have a lot of floating boxes that highlight important information for kids, including: 

• Pointing out various strategies they can take on given concepts or problems
• Offering extra work
• Giving extra information
• Even offering “bogus” solutions that point out the most common mistakes made by students when solving a problem

Despite its rigor, Art of Problem Solving does its best to keep its material from becoming too dry and boring, which we appreciate. 

The books are written in a very casual tone, which makes it feel as if a math-whiz friend were explaining the material rather than a textbook. 

There are also a good deal of amusing and interesting examples and concept demonstrations sprinkled throughout, sometimes even involving sly pop culture references (some of which may go over kids heads, but parents will appreciate).

Does this approach really work?

Due to its philosophy and the way it teaches, we feel the Art of Problem solving takes more or less a constructivist/Problem Based Learning approach to teaching math where, instead of receiving formal lectures about math, students build up their own knowledge and skill by working through and solving various problems.

This learner-centric approach to teaching math and science actually has been linked to positive outcomes when teaching math and science , fostering greater problem-solving skills, improving self-motivation and encouraging creative and critical thinking skills as they relate to mathematics. 

Past customers have also reported that the series challenges their students pretty thoroughly, increasing the depth of their knowledge on relevant subjects and increasing their speed at solving difficult-math problems, sometimes dramatically. 

It is perhaps unsurprising, then, that the Art of Problem Solving curriculum is often used in honors math classes across the US. 

Some Drawbacks to Art of Problem Solving Textbooks and Curriculum

Can be time consuming.

Due to its focus on doing exercises, exploring concepts and working through problems to gain a better understanding of the subject matter, Art of Problem Solving can take a little more time to work with than some other programs.

This can be particularly true as AoPS tends to use far more challenging questions than kids are used to, some of which are in formats they haven’t seen before. 

While great for learning, this approach isn’t exactly a time saver. It’s not uncommon, for example, for parents to report spending up to 45 min (or more) each day on math (in addition to other homework). 

Can be tricky to jump into from another curriculum

With its particular approach and pedagogy, as well as its more rigorous approach to mathematics and problem solving (including the use of proofs), Art of Problem Solving can be somewhat tricky to get used to if you jump into it from another curriculum. 

Because math is a cumulative process, kids who begin Art of Problem solving without having at least reviewed some of the foundational material in previous books can find themselves lost or slowed down by skill and knowledge gaps they didn’t realize they had. 

Helpfully, the AoPS website does have free, printable diagnostic assessments for each book to help parents determine if their kids are at the right skill level. 

Discovery approach can frustrate some learners

Despite the fact that Age of Problem Solving’s approach has been shown to get results and improve the mathematical thinking and skills of talented math students, sometimes it just isn’t the right approach for the student. 

AoPS often requires students to play around with numbers and concepts and discover missing information themselves. 

Some students, even really talented students, can get frustrated by this approach and may prefer a more straightforward, traditional math course where they can get down to computation and see their results more quickly. 

Who is Art of Problem Solving For?

Overall, we think Art of Problem Solving is a great resource for parents and kids looking for a far more thorough, challenging and enriched math program.

It is an ideal course for students who demonstrate an aptitude for math and are looking to deepen and strengthen their math skills with more challenging grade-level material.

We think AoPS textbooks can be particularly good for students interested for more rigorous preparation for math-heavy STEM subjects in university , where their greater focus on problem solving, proofs and logic skills will be a strong asset, such as with physics, engineering and even computer science,

We also think that Art of Problem solving’s textbooks and methodology can be an excellent base material for students interested in or preparing for math contests and olympiads (AMC 10, AMC 12, MATHCOUNTS and the like), particularly their Contest Math Prep Series, as they promote creative approaches to problem solving and strengthen mathematical thinking that kids can use when faced with new problems.

Who is Art of Problem Solving Not Great For?

That said, Art of Problem Solving textbooks are obviously not for every student. 

These books are not the best curriculum for kids who are struggling with math concepts as AoPS math is primarily aimed at enriching math study. 

AoPS math goes far deeper into the material with far more rigor, exploring various high school and middle school math topics at a more advanced level and with more challenging problem sets, while emphasizing multiple approaches to problem solving and flexibility when approaching new math problems. 

Struggling students, while they often can benefit from learning the why’s behind math, can usually spend their time better by reviewing the fundamentals and practicing basic strategies, as well as by working on more targeted skill development with programs like IXL and Khan Academy .

Similarly, we don’t feel that AoPS textbooks are really the best resource for preparing for the SAT and other timed standardized tests where answering speed and efficiency (and test taking strategies) can be far more effective when it comes to success than gaining a deep understanding of concepts and working through problems.

In these instances, kids are better served through specific standardized prep programs that will work with them on developing their proficiency at solving very particular types of questions. 

Finally, AoPS textbooks are also not the best solution for kids looking to explore college level math as, despite its more challenging nature, AoPS math goes deeper into middle school and high school math topics (algebra, geometry, number theory, single variable calculus), rather than beyond it.

Price: How much do AoPS Textbooks Cost? 

The price of AoPS math textbooks really depends on the particular book and subject you’re interested in. 

Generally speaking, though, each book costs between $45 and $70, which is roughly the same as the average middle or high school textbook .

The length of each book varies, however, from just under 300 pages of instructional material in some cases to well over 700 in others.

Unlike many other middle and high school textbooks, however, these are designed to serve as a complete curriculum for each topic as every book contains instructional material as well as hundreds of practice problems, hints, and a step-by-step solution guide that itself is usually a couple hundred pages long as well. 

Bottom Line:

If you have a talented middle or high school math student and you’re looking for ways to nurture their excellence, Art of Problem Solving’s math textbooks might be right for you.

Although certainly not for everyone, with their challenging curriculum and in-depth exploration of math concepts, AoPS can foster better problem solving skills, stronger analytical ability and improved creative and critical math thinking, all of which can help students take their math skills to the next level.

About the Author

David Belenky is a freelance writer, former science and math tutor and a tech enthusiast. When he’s not writing about educational tech, he likes to chill out with his family and dog at home.  

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The art of problem solving 7th edition

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Rigorous Curriculum and an Unforgettable Experience

Since 1993, Art of Problem Solving has prepared hundreds of thousands of motivated students in grades 2–12 for college and career success. Through our innovative approach, students build a problem-solving foundation, an unparalleled skill set that helps them overcome obstacles in school and in life.

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Our curriculum is rigorous — it's built to be. If students never feel challenged, they’re not getting to the boundaries of what’s possible. By solving new and complex problems, AoPS students are inspired to expand to their fullest academic potential.

If you’re looking for a challenging, interactive environment where your student will build the skill stack to succeed in school and beyond, you'll find it at AoPS Academy Princeton!

Our family regularly talks about AoPS. We try to think about how our life would be different without you all. What if my daughter hadn't learned to love math? What if she never experienced being pushed to her limits? Overcoming failure? She wouldn't be who she is. And she is AWESOME! A very proud mom here. We are so fortunate that we found AoPS Academy. I credit AoPS for much of her confidence.

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Art of Problem Solving: Introducing Ratios

TLDR This educational video script introduces the concept of ratios through relatable scenarios, such as kids to adults at a party. It explains the meaning of a ratio, using the example of a party with a 5:2 kids to adults ratio, and demonstrates how to calculate the number of kids when given the number of adults. The script also explores another ratio problem with a 3:5 kids to adults ratio and shows how to determine the number of kids when there are 26 more adults. The video uses both direct multiplication of the ratio and algebraic methods to solve the problems, reinforcing the concept of equivalent ratios.

• 🎉 The concept of ratios is introduced as a way to compare quantities, specifically kids to adults at a party.
• 🔍 A ratio is expressed as '5 to 2', meaning for every 5 kids, there are 2 adults, without specifying the total number of people.
• 👶 The ratio can be scaled up by multiplying both parts by the same number to find an equivalent ratio that fits a given scenario.
• 🧮 If 14 adults are at a party with a '5 to 2' kids to adult ratio, there would be 35 kids, found by multiplying both parts of the ratio by 7.
• 🤔 The ratio helps to determine the number of kids when the number of adults is known, by maintaining the proportionality.
• 🎈 Another example ratio, '3 to 5' kids to adults, suggests a different party composition, implying fewer kids relative to adults.
• 🧐 The problem-solving approach involves creating groups based on the ratio and scaling these groups to match given conditions.
• 🔢 For a '3 to 5' ratio with 26 more adults than kids, 13 groups are needed, leading to 39 kids and 65 adults.
• 📚 The script illustrates two methods to solve ratio problems: direct multiplication of the ratio and using variables (x) to represent groups.
• 🎓 Understanding ratios is crucial for determining the number of individuals in one category based on the number in another, given their ratio.
• 💡 The script emphasizes the importance of ratios in real-life situations like parties, where the balance between different groups can affect the outcome.

What is the primary concern for a child when considering whether to attend a party?

- The primary concern for a child is to determine if there will be more kids than adults at the party, as this would indicate whether it's a kids party or an adults party.

What does the ratio of kids to adults represent in the context of the party?

- The ratio of kids to adults represents the proportion of kids to adults at the party. For example, a ratio of 5 to 2 means for every 5 kids, there are 2 adults.

How does the ratio help in determining the number of kids at a party if the number of adults is known?

- The ratio helps by setting up a proportional relationship between the number of kids and adults. If the ratio is known and the number of adults is given, you can scale the ratio to match the given number of adults and then calculate the number of kids accordingly.

What is the ratio of kids to adults at the first party problem described in the transcript?

- The ratio of kids to adults at the first party problem is 5 to 2.

If there are 14 adults at the party with a ratio of 5 kids to 2 adults, how many kids are there?

- With a ratio of 5 kids to 2 adults, and 14 adults at the party, there would be 35 kids at the party.

How can you scale a ratio to match a given number of adults?

- You can scale a ratio to match a given number of adults by multiplying both parts of the ratio by the same number until the number of adults in the scaled ratio matches the given number.

What is an equivalent ratio and how is it used in solving the party problem?

- An equivalent ratio is a ratio that has the same relationship between its parts as the original ratio but with different numbers. It is used in solving the party problem by scaling the original ratio to match the given number of adults, thus allowing the calculation of the number of kids.

In the second party problem, what is the ratio of kids to adults and what does it imply about the party?

- In the second party problem, the ratio of kids to adults is 3 to 5, which implies that for every 3 kids, there are 5 adults, suggesting it might be a less kid-friendly party.

How can you determine the number of kids at a party if there are 26 more adults than kids and the ratio is 3 kids to 5 adults?

- If there are 26 more adults than kids and the ratio is 3 kids to 5 adults, you can determine the number of kids by setting up an equation where 5x (the number of adults) is 26 more than 3x (the number of kids), solving for x, and then calculating 3x.

What is the significance of the number 13 in the second party problem?

- In the second party problem, the number 13 represents the number of groups of 3 kids and 5 adults needed to have a total of 26 more adults than kids at the party.

🎉 Understanding Party Ratios

This paragraph introduces the concept of ratios in the context of children's parties. It explains that children are more interested in the ratio of kids to adults at a party rather than the total number of attendees. The script uses a problem-solving approach to teach ratios, starting with a 5:2 kids to adults ratio at a party. It clarifies that this ratio does not indicate the total number of people but the proportion of kids to adults. The paragraph then solves a problem where, given 14 adults, it calculates the number of kids at the party by scaling the ratio to match the given number of adults. Two methods are presented: directly multiplying the ratio parts by a number that aligns with the given adults, and using a variable 'x' to represent the number of groups, leading to the same conclusion that there are 35 kids at the party.

🧮 Exploring Different Ratio Scenarios

The second paragraph delves into another ratio problem, this time with a 3:5 kids to adults ratio, suggesting a less child-friendly party. The paragraph poses a scenario where there are 26 more adults than kids and uses both direct multiplication and algebraic methods to find the number of kids. The direct method involves multiplying the ratio by a number that results in 26 more adults than kids, concluding with 13 groups that yield 39 kids. The algebraic method sets up an equation based on the ratio and the given difference, solving for 'x' to find the same result of 39 kids. The paragraph emphasizes the importance of understanding ratios and demonstrates that different methods can be used to arrive at the same solution.

💡 Multiplication

💡 equivalent ratio, 💡 problem solving, 💡 variables, 💡 equations, 💡 difference.

Understanding the concept of ratios through the context of a kids' party.

The ratio of kids to adults at a party is described as '5 to 2'.

A ratio does not specify the total number of people, only the relationship between two groups.

Groups at a party can be formed based on the ratio, with each group containing 5 kids and 2 adults.

Multiplying the components of a ratio by the same number results in an equivalent ratio.

Using the ratio to determine the number of kids when given the number of adults.

If there are 14 adults at the party, there must be 35 kids, maintaining the 5:2 ratio.

Exploring another ratio problem with a 3 to 5 kids to adults ratio.

A party with a 3:5 ratio suggests a higher number of adults, potentially making it less exciting for kids.

Determining the number of kids when there are 26 more adults than kids.

Using the ratio to calculate that there are 39 kids when there are 13 groups of 3 kids and 5 adults.

Verifying the solution by checking that 65 adults are indeed 26 more than 39 kids.

Introducing the variable 'x' to represent the number of groups in the ratio.

Solving the problem algebraically by setting up the equation 5x - 3x = 26.

Finding that x equals 13, which confirms the earlier conclusion of 39 kids at the party.

Emphasizing the importance of understanding ratios in problem-solving.

Highlighting that the same problem can be solved in multiple ways, ensuring the solution's validity.

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