problem solving theory author

Problem Solving

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• pp 2680–2683
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• David H. Jonassen 2 &
• Woei Hung 3  

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Cognition ; Problem typology ; Problem-based learning ; Problems ; Reasoning

Problem solving is the process of constructing and applying mental representations of problems to finding solutions to those problems that are encountered in nearly every context.

Theoretical Background

Problem solving is the process of articulating solutions to problems. Problems have two critical attributes. First, a problem is an unknown in some context. That is, there is a situation in which there is something that is unknown (the difference between a goal state and a current state). Those situations vary from algorithmic math problems to vexing and complex social problems, such as violence in society (see Problem Typology ). Second, finding or solving for the unknown must have some social, cultural, or intellectual value. That is, someone believes that it is worth finding the unknown. If no one perceives an unknown or a need to determine an unknown, there is no perceived problem. Finding...

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Bransford, J., & Stein, B. S. (1984). The IDEAL problem solver: A guide for improving thinking, learning, and creativity . New York: WH Freeman.

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Newell, A., & Simon, H. (1972). Human problem solving . Englewood Cliffs: Prentice Hall.

Rumelhart, D. E., & Norman, D. A. (1988). Representation in memory. In R. C. Atkinson, R. J. Herrnstein, G. Lindzey, & R. D. Luce (Eds.), Steven’s handbook of experimental psychology (Learning and cognition 2nd ed., Vol. 2, pp. 511–587). New York: Wiley.

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School of Information Science and Learning Technologies, University of Missouri, 221C Townsend Hall, 65211, Columbia, MO, USA

Dr. David H. Jonassen

College of Education and Human Development, University of North Dakota, 231 Centennial Drive, Stop 7189, 58202, Grand Forks, ND, USA

Dr. Woei Hung

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Correspondence to David H. Jonassen .

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Faculty of Economics and Behavioral Sciences, Department of Education, University of Freiburg, 79085, Freiburg, Germany

Norbert M. Seel

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Jonassen, D.H., Hung, W. (2012). Problem Solving. In: Seel, N.M. (eds) Encyclopedia of the Sciences of Learning. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1428-6_208

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12 Best Problem Solving Books to Read

By: Angela Robinson | Updated: June 30, 2023

You found our list of top problem solving books .

Problem solving books are guides that improve critical thinking capability and the ability to resolve issues in the workplace. These works cover topics like bias and logical fallacies, problem prevention, and prioritizing. The purpose of these books is to help workers remain calm under pressure and come up with solutions more quickly.

These guides are similar to decision making books , negotiation books , and conflict resolution books . To improve competency in this area, one can also play problem solving games .

This list includes:

• problem solving books for adults
• creative problem solving books
• business problem solving books
• problem solving books for programmers

Here we go!

List of problem solving books

Here is a list of books to improve problem solving skills in the workplace.

1. Fixed: How to Perfect the Fine Art of Problem Solving by Amy E Herman

Fixed is one of the most useful new books on problem solving. The book calls for problem solvers to look beyond instinctual and obvious answers and provides a framework for more creative thinking. While most folks think about problem solving in terms of logic, reason, and disciplines like math and science, this book shows the role that art and imagination play in the process. Amy Herman consulted on leadership training with Silicon Valley companies and military organizations and brings this expertise into the text to train readers on how to adopt a more innovative critical thinking approach.

Notable Quote: “Working through problems is critical for productivity, profit, and peace. Our problem-solving skills, however, have been short-circuited by our complicated, technology-reliant world.”

Read Fixed .

2. Cracked it!: How to solve big problems and sell solutions like top strategy consultants by Bernard Garrette, Corey Phelps, and Olivier Sibony

Cracked it! is one of the best creative problem solving books. Drawing inspiration from the tactics of consultants, this guide is a practical playbook for approaching business problems. The authors outline a “4S” method– State – Structure – Solve – Sell– to tackle obstacles and get support from stakeholders. While many problem solving books simply focus on how to think through issues, this guide also demonstrates how to gain approval for ideas and get others onboard with the solution. The book explains how to best use these techniques, and presents case studies that show the theories in action. Cracked it! is a handy reference for any professional that faces tough challenges on the regular.

Notable Quote: “If you want to know how a lion hunts, don’t go to a zoo. Go to the jungle.”

Read Cracked it!

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3. Upstream: The Quest to Solve Problems Before They Happen by Dan Heath

Upstream takes a proactive approach to problem solving. The book urges readers to not only be responsive to issues, but also try to prevent obstacles from occurring. The guide opens with an exploration of “problem blindness,” and the psychological factors that cause folks to be oblivious to issues, along with a reminder that many problems are more controllable and avoidable than first assumed. The pages that follow outline a series of questions leaders can ask to fine-tune the system and steer clear of major headaches, for instance, “How Will You Unite the Right People?” and “How Will You Avoid Doing Harm?” Upstream is full of real world examples of how minor tweaks achieved major results and allowed organizations to sidestep serious holdups.

Notable Quote: “The postmortem for a problem can be the preamble to a solution.”

Read Upstream .

4. Problem Solving 101: A Simple Book for Smart People by Ken Watanabe

Problem Solving 101 is one of the most fun problem solving books for adults. Written by Ken Watanabe, the guide draws on Japanese philosophy as well as the author’s experience as a consultant at McKinsey to help readers understand and approach problems in productive ways. The pages provide blueprints for problem-solving methods such as logic trees and matrixes, and include scenarios and illustrations that help readers visualize the process more clearly. Problem Solving 101 breaks down the problem solving procedure into the most basic parts and lays out step-by-step instructions for choosing the best action in any situation.

Notable Quote: “When you do take action, every result is an opportunity to reflect and learn valuable lessons. Even if what you take away from your assessment seems to be of small consequence, all of these small improvements taken together make a huge difference in the long term.”

Read Problem Solving 101 .

5. What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve by Thomas Wedell-Wedellsborg

What’s Your Problem? insists that the most important step in the problem solving process is to start by honing in on the correct problem. The root of much frustration and wasted efforts is that professionals often pick the wrong points to focus on. This book teaches readers how to reframe and approach issues from a different perspective. The guide outlines a repeatable three step process “Frame, Reframe, and Move Forward” to ensure that workers prioritize effectively and stay on track to achieve desired results. What’s Your Problem? teaches professionals of all levels how to be less rigid and more results-focused and adopt a more agile approach to fixing issues.

Notable Quote: “The problems we’re trained on in school are often quite different from the ones we encounter in real life.”

Read What’s Your Problem?

6. Sprint: How to Solve Big Problems and Test New Ideas in Just Five Days by Jake Knapp, John Zeratsky, et al

Sprint is one of the best problem solving books for programmers. The authors are the creators of the five-day-process at Google. This guide describes best practices for conducting sprints and solving problems in limited timeframes. The book provides a day-by-day breakdown of tasks for each day of the workweek, with the final steps being designing a prototype and a plan for implementation. Though this idea originated in the tech world and is most widely used in the software industry, this problem-solving and product design approach can be useful for any position that needs to find fixes in a time crunch.

Notable Quote: “We’ve found that magic happens when we use big whiteboards to solve problems. As humans, our short-term memory is not all that good, but our spatial memory is awesome. A sprint room, plastered with notes, diagrams, printouts, and more, takes advantage of that spatial memory. The room itself becomes a sort of shared brain for the team.”

Read Sprint , and check out this guide to virtual hackathons and this list of product design books .

7. Think Like a Rocket Scientist: Simple Strategies You Can Use to Make Giant Leaps in Work and Life by Ozan Varol

Think Like a Rocket Scientist lays out formulas and instructions for thinking more strategically. The guide reveals common problem solving approaches used by rocket scientists when exploring the unknown and testing new technology. The book is split into three sections– launch, accelerate, and achieve– with deep dives into concepts such as moonshot thinking and overcoming failure. The anecdotes revolve around space exploration and rocket science yet the methods can be applied to more commonplace and less complex problems as well. Think Like a Rocket Scientist proves that one does not need to be a genius to be a genius problem solver and lets readers learn tricks from one of the most complex professions on the planet.

Notable Quote: “Critical thinking and creativity don’t come naturally to us. We’re hesitant to think big, reluctant to dance with uncertainty, and afraid of failure. These were necessary during the Paleolithic Period, keeping us safe from poisonous foods and predators. But here in the information age, they’re bugs.”

Read Think Like a Rocket Scientist .

8. Bulletproof Problem Solving: The One Skill That Changes Everything by Charles Conn and Robert McLean

Bulletproof Problem Solving is one of the best business problem solving books. This workbook-style-guide breaks down a “bulletproof” method of problem solving favored by consultants at McKinsey. The authors distill the process into seven simple steps–define the problem, disaggregate, prioritize, workplan, analyze, synthesize, and communicate– and give numerous examples of how to follow this cycle with different dilemmas. The chapters explore each stage in depth and outline the importance and finer points of each phase. The book also provides practical tools for readers to build skills, including an appendix with exercise worksheets.

Notable Quote: “Problem solving doesn’t stop at the point of reaching conclusions from individual analyses. Findings have to be assembled into a logical structure to test validity and then synthesized in a way that convinces others that you have a good solution. Great team processes are also important at this stage.”

Read Bulletproof Problem Solving .

9. Think Like a Programmer: An Introduction to Creative Problem Solving by by V. Anton Spraul

Think Like a Programmer is one of the top problem solving books for programmers. The guide lays out methods for finding and fixing bugs and creating clean, workable code. The text emphasizes that programming is not merely a matter of being competent in the language, but also knowing how to troubleshoot and respond to unexpected occurrences. The chapters present examples of problems and puzzles and work through the answers to help strengthen professional competencies. The book provides an introductory crash course and practical toolkit for beginning coders, with a focus on C++. Yet since the text outlines general theory and approach, the book is also helpful for dealing with other programming languages, or for solving problems in non-tech industries as well. The point of the text is to provide a proper mindset and attitude for reacting to these developments, and the book can be a benefit for folks in any field.

Notable Quote: “Don’t Get Frustrated The final technique isn’t so much a technique, but a maxim: Don’t get frustrated. When you are frustrated, you won’t think as clearly, you won’t work as efficiently, and everything will take longer and seem harder. Even worse, frustration tends to feed on itself, so that what begins as mild irritation ends as outright anger.”

Read Think Like a Programmer .

10. The Founder’s Dilemmas: Anticipating and Avoiding the Pitfalls That Can Sink a Startup by by Noam Wasserman

The Founder’s Dilemmas lays out the most common problems entrepreneurs face and gives advice on how to avoid or solve these issues. The book tackles topics such as managing relationships, hiring, and rewarding or correcting employees. The chapters outline the mistakes inexperienced leaders often make and offer strategies for handling these tough situations with more smarts and skill. By reading this book, founders can learn from predecessors and avoid making obvious and avoidable errors in judgment. The Founder’s Dilemmas is a problem-solving resource for startup leaders and team members who lack more traditional guidance.

Notable Quote: “Ideas are cheap; execution is dear.”

Read The Founder’s Dilemmas , and check out more entrepreneurial books .

11. The Scout Mindset: Why Some People See Things Clearly and Others Don’t by Julia Galef

The Scout Mindset challenges readers to move beyond gut reactions and preconceptions and rethink problems. The book offers instructions for overcoming bias and central beliefs to gather more objective data. Julia Galef encourages readers to act more like scouts than soldiers and gather information without judging to make more informed decisions. The text outlines the common reasons folks jump to conclusions and offers advice on how to avoid incorrect assumptions and conduct level-headed analyses. The Scout Mindset is a call to action for objectivity and an instruction manual for breaking away from unhelpful mental patterns that can lead to poor choices.

Notable Quote: “Discovering you were wrong is an update, not a failure, and your worldview is a living document meant to be revised.”

Read The Scout Mindset .

12. Super Thinking: The Big Book of Mental Models by Gabriel Weinberg and Lauren McCann

Super Thinking is a comprehensive resource that explains various mental models for problem solving. The book identifies logical fallacies and shows readers how to avoid these pitfalls. The pages also lay out appropriate strategies, tools, techniques to use in different situations, such as matrices, pointed questions, and philosophies. The point of the guide is to teach readers how to evaluate information and make quick yet accurate judgements. The guide helps readers decide the best approach to use for each circumstance. Though packed with information, the pages also contain images and humor that prevent the material from getting too dry. Super Thinking is the ultimate cheat sheet for thinking rationally and acting with intention.

Notable Quote: “Unfortunately, people often make the mistake of doing way too much work before testing assumptions in the real world.”

Read Super Thinking .

Final Thoughts

Problem solving is one of the most essential skills for modern industry. With the breakneck pace at which the current business world changes, there is no shortage of new developments that professionals must contend with on a daily basis. Operating the same way for years at a time is impossible, and it is almost guaranteed that workers at every level will have issues to unravel at some point in their careers.

Books about problem solving help professionals predict, prevent, and overcome issues and find more viable and sustainable solutions. These guides not only provide skills, but also methods for survival in a highly competitive business landscape. These texts show workers that they are more capable than may first appear and that sometimes, seemingly insurmountable obstacles are beatable with a combination of creativity, teamwork, and proper process.

For more ways to beat the odds, check out this list of books on innovation and this list of books on business strategy .

We also have a list of the best communication books .

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FAQ: Problem solving books

Here are answers to common questions about problem solving books.

What are problem solving books?

Problem solving books are guides that teach critical thinking skills and strategies for resolving issues. The purpose of these works is to help professionals be more creative and strategic in problem solving approaches.

What are some good problem solving books for work?

Some good problem solving books for work include Sprint by Jake Knapp, John Zeratsky, et al, Upstream by Dan Heath, and Think Like a Rocket Scientist by Ozan Varol.

Author: Angela Robinson

Marketing Coordinator at teambuilding.com. Angela has a Master of Fine Arts in Creative Writing and worked as a community manager with Yelp to plan events for businesses.

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Problem-solving concepts and theories

Affiliation.

• 1 Mississippi StateUniversity, College of Veterinary Medicine, USA. [email protected]
• PMID: 14648495
• DOI: 10.3138/jvme.30.3.226

Many educators, especially those involved in professional curricula, are interested in problem solving and in how to support students' development into successful problem solvers. The following article serves as an overview of educational research on problem solving. Several concepts are defined and the transition from one theory to another is discussed. Educational theories describing problem solving in the context of behavioral, cognitive, and information-processing pedagogy are discussed. The final section of the article describes prior findings regarding expert-novice differences in problem solving of various kinds.

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Olympiad books

Here is a list of Olympiad Books that have Olympiad-level problems used to train students for future mathematics competitions.

You can discuss here about these books or request new books. Let's categorize books into Theory books, Problem books, and Both books.

• 1.1 Inequalities
• 1.2 Polynomials
• 1.3 Functional Equations
• 2 Number Theory
• 4 Combinatorics
• 5.1 Algebra
• 5.2 Number Theory
• 5.3 Geometry
• 5.4 Combinatorics
• 5.5 General Problem Solving

Inequalities

• Advanced Olympiad Inequalities: Algebraic & Geometric Olympiad Inequalities by Alijadallah Belabess .
• Inequalities An Approach Through Problems - B. J. Venkatachala
• Secrets In Inequalities volume 1 - Basic Inequalities - Pham Kim Hung .
• Secrets In Inequalities volume 2 - Advanced Inequalities - Pham Kim Hung .
• Algebraic Inequalities - Old And New Methods - Vasile Cirtoaje .
• Old And New inequalities volume 1 - Titu Andreescu, Vasile Cirtoaje, Gabriel Dospinescu, Mircea Lascu .
• Old And New Inequalities volume 2 - Vo Quoc Ba Can, Cosmin Pohoata .
• The Cauchy-Schwarz Master Class - J. Michael Steele .
• Inequalities: A Mathematical Olympiad Approach - Radmila Bulajich Manfrino, Jose Antonio Ortega, Rogelio Valdez Delgado .
• An Introduction to Inequalities - Bellman, Beckenbach .
• Analytic Inequalities - Mitrinovic .
• Inequalities Theorems and Formulas forum .
• Useful Inequalities topic.

Polynomials

• Olympiad Algebra Book (Vol I): 1220 Polynomials & Trigonometry Problems by Amir Parvardi .
• Awesome Polynomials for Mathematics Competitions (XYZ-Press)- Titu Andreescu-Navid Safaei-Alessandro Ventullo .
• 117 Polynomial Problems from the Awesomemath Summer Program -(XYZ-Press)- Titu Andreescu-Navid Safaei-Alessandro Ventullo .
• Polynomials and Polynomial Inequalities (Graduate Texts in Mathematics) - (Springer) - Peter Borwein - Tamas Erdely .
• Geometry of Polynomials - (American Mathematical Society) - Morris Marden .
• Polynomials - (Springer) - E.J. Barbeau .
• Solving Polynomial Equations: Foundations, Algorithms, and Applications - (Springer) - Alicia Dickenstein - Ioannis Z. Emiris .

Functional Equations

• Topics in Functional Equations: Third Edition - Titu Andreescu, Iurie Boreico, Oleg Mushkarov, Nikolai Nikolov .
• Functional Equations in Mathematical Olympiads (2017 - 2018): Problems and Solutions (Vol. I) - (Amazon) - Amir Hossein Parvardi .
• Functional Equations and Inequalities in Several Variables - (World Scientific Publication) - Stefan Czerwik .
• Lectures on Functional Equations - (Academic Press) - J. Aczel .
• Functional Equations: A Problem Solving Approach - (Prism Books) - B.J. Venkatchala .
• Functional Equations and How to Solve Them - (Springer) - Christopher G. Small .

Number Theory

• Number Theory Structures, Examples, and Problems - Titu Andreescu, Dorin Andrica - Both Book (olympiad examples followed by problems). Excellent book for number theory.
• Number Theory: Concepts and Problems - Gabriel Dospinescu, Oleg Mushkarov, and Titu Andreescu - Both Book (olympiad examples followed by problems). Excellent book for number theory, and has it's own unique approach, Highly Suggested for Number Theory.
• An Introduction to Diophantine Equations - Titu Andreescu, Dorin Andrica, Ion Cucurezeanu - Both Book (olympiad examples followed by problems). Excellent book for Diophantine equations.
• 104 Number Theory Problems - Titu Andreescu, Dorin Andrica, Zuming Feng - Both Book.
• 250 problems in number theory - W. Sierpinski - Problem Book.
• Modern Olympiad Number Theory - Aditya Khurmi - Both Book.
• A Selection of Problems in Theory of Numbers - W. Sierpinski - Problem Book. Great book.
• The Theory of Numbers - a Text and Source Book of Problems - Andrew Adler, John E. Coury - Both Book (olympiad examples followed by problems). Excellent book.
• Number Theory - Naoki Sato (nsato) - Theory Book.
• Solved and Unsolved Problems in Number Theory - Daniel Shanks - Problem Book.
• Elementary Number Theory (Revised Printing) - David M. Burton - It is a nice book for theory building and is low-impact in its approach.
• An Introduction to the Theory of Numbers - Ivan Niven, Herbert S. Zuckerman - Theory Book.
• Elementary Number Theory - W. Edwin Clark - Theory Book.
• Numbers and Curves - Franz Lemmermeyer - Theory Book.
• Algorithmic Number Theory - S. Arun-Kumar - Theory Book.
• Elementary Number Theory - William Stein - Both Book (lots of theorems with problems at the end of each section).
• Number Theory, An Introduction via the Distribution of Primes - Benjamin Fine, Gerhard Rosenberger - Theory Book.
• Number Theory for Computing - Song Y. Yan - Theory Book (this book contains computational examples/theorems for number theory).
• Pell's Equation - Edward J. Barbeau [level is a little above olympiad] - Both Book (olympiad examples followed by problems).
• Topics in Number Theory - Masum Bilal and Amir Hossein Parvardi - Both Book
• 103 Trigonometry Problems - Titu Andreescu, Zuming Feng - Both book (solved examples and approaches + problems).
• Triangles, Concurrency and Quadrilaterals
• Geometry Unbound - Kedlaya - Theory book - this book is available online for download. See here l - Great book.
• Famous Problems of Geometry and How to Solve Them - Benjamin Bold - Both book (solved examples and approaches + problems).
• Challenging Problems in Geometry - Alfred S. Posamenter, Charles T. Salkind - Both book - Great book.
• Euclidean Geometry in Mathematical Olympiads - Evan Chen - Both book - good book. By far the greatest geometry book to prepare for olympiads. if you had to choose one book, its definitely this one
• Elements of Projective Geometry - Luigi Ceremona - Both book, again.
• Japanese Temple Geometry Problems - San Gaku - Problem book (it contains lots of theorems about circles).
• Geometric Problems on Maxima and Minima - Titu Andreescu, Oleg Mushkarov, Luchezar Stoyanov - Problem book - Great book.
• Complex Numbers in Geometry - I. M. Yaglom - Theory book.
• All Volumes (direct link to the RAR file) - Alternative link: All Volumes TOTO SLOT
• Darij Grinberg's whole site download - his website has a great number of articles/solved problems that you may use in your Olympiad studying - Great .
• Geometry revisited - Coxeter and Greitzer - Both book.
• Problems in Geometry - Kutepov, Rubanov - Problem book.
• Investigations in Geometry (Math Motivators!) - Posamentier, Sheridan - Both book.
• Introduction to Geometry - Coxeter - Theory book.
• Modern Geometry with Applications - Jennings - Both book.
• Geometric Transformations (4 volumes) - Yaglom - Theory book.

Combinatorics

• A Path to Combinatorics for Undergraduates - Andreescu, Feng .
• Proofs that Really Count (The Art of Combinatorial Proof)' - Benjamin and Quinn .
• A Course in Combinatorics - Lint and Wilson .
• Olympiad Combinatorics - Pranav A. Sriram .

Improve Your Skills With Problem Solving

• 100 Inequality Problems Proposed by Vasc and arqady - Amir Hossein Parvardi .
• 116 Algebra Problems - Mohammad Jafari .
• 100 Nice Polynomial Problems (With Solutions) - Amir Hossein Parvardi .
• 100 Functional Equations Problems (With Solutions) - Amir Hossein Parvardi .
• Trigonometry Problems Collection - Amir Hossein Parvardi .
• 567 Nice And Hard Inequalities - Nguyễn Duy Tùng .
• Inequalities From 2007 and 2008 Competitions Around The World - Manh Dung Nguyen .
• Technical Analysis of Three Variable Inequalities - Nguyen Duy Tung, Zhou Yuan Zhe .
• 1220 Number Theory Problems (With Sources) - Amir Hossein Parvardi .
• Lifting the Exponent Lemma (LTE) - Amir Hossein Parvardi .
• Solving Diophantine Equations - lamphong .
• Several Things About Sum of Squares - lamphong .
• Some Own Problems In Number Theory - mathmdmb .
• Primitive Roots, Order, and Quadratic Residues - mathmdmb .
• Number Theory Marathon Problems - M4RI0 .
• 150 Nice Geometry Problems (With Solutions) - Amir Hossein Parvardi .
• Darij Grinberg's whole site download - his website has a great number of articles/solved problems that you may use in your Olympiad studying.

Euclidean Geometry in Mathematical Olympiads [1]

• 100 Combinatorics Problems (With Sources) - Amir Hossein Parvardi .
• 102 Combinatorial Problems - Andreescu, Feng .
• Problems in Combinatorics and Graph Theory - Ioan Tomescu .

General Problem Solving

• Challenging Mathematical Problems With Elementary Solutions (Volume I, Combinatorial Analysis and Probability Theory) - A. M. Yaglom, I. M. Yaglom .
• Challenging Mathematical Problems With Elementary Solutions (Volume II, Problem From Various Branches of Mathematics) - A. M. Yaglom, I. M. Yaglom .
• AoPS Resources Page Problems (IMO and ShortLists Added) - Amir Hossein Parvardi .
• Mathematics as Problem Solving - Alexander Soifer .
• A Primer For Mathematics Competitions - Alexander Zawaira, Gavin Hitchcock .
• Problem Solving Strategies For Efficient And Elegant Solutions (A Resource For The Mathematics Teacher) - Alfred S. Posamentier, Stephen Kruli .
• Problems for the Mathematical Olympiads (From the First Team Selection Test to the IMO) - Andrei Negut .
• Problem Primer for the Olympiad - C. R. Pranesachar, B. J. Venkatachala, C. S. Yogananda .
• Chinese Mathematics Competitions and Olympiads (two volumes) - Andy Liu .
• Hungarian Problem Book' (three volumes) - Andy Liu .
• Canadian Mathematical Olympiad 1969-1993 (Problems and Solutions) - Michael Doob .
• The Art and Craft of Problem Solving - Paul Zeitz .
• APMO 1989-2009 (Problems & Solutions) - Dong Suugaku - download here .
• International Mathematical Olympiads 1978-1985 and Forty Supplementary Problems - Murray S. Klamkin .
• USA Mathematical Olympiads 1972-1986 (Problems and Solutions) - Murray S. Klamkin .
• USSR Mathematical Olympiads 1989-1992 - Arkadii M. Slinko .
• Proofs From THE BOOK - Martin Aigner, Günter M. Ziegler .
• Techniques of Problem Solving - Steven G. Krantz .
• Junior Balkan Mathematical Olympiads - Dan Branzei, loan Serdean, Vasile Serdean .
• The IMO Compendium (A Collection of Problems Suggested for the Mathematical Olympiads, 1959-2004) - Dusan Djukic, Vladimir Jankovic, Ivan Matic, Nikola Petrovic .
• Five Hundred Mathematical Challenges - Edward J. Barbeau, Murray S. Klamkin, William O. J. Moser .
• The USSR Olympiad Problem Book (Selected Problems and Theorems of Elementary Mathematics) - D. O. Shklarsky, N. N. Chentzov, I. M. Yaglom .
• The William Lowell Putnam Mathematical Competition (Problems and Solutions 1965-1984) (three volumes) - Volume 1 : A. M. Gleason, R. E . Greenwood, L. M. Kelly, Volume 2 : Gerald L. Alexanderson, Leonard F. Klosinski, Loren C. Larson , Volume 3 : Kiran S. Kedlaya, Bjorn Poonen, Ravi Vakil .
• International Mathematics TOURNAMENT OF THE TOWNS (Questions & Solutions) - (five volumes) - Peter J. Taylor .
• Mathematical Problems and Proofs (Combinatorics, Number Theory and Geometry) - Branislav Kisacanin .
• 360 Problems for Mathematical Contests - Titu Andreescu, Dorin Andrica .
• PROBLEMS FROM AROUND THE WORLD - (six volumes) - Titu Andreescu, Kiran S. Kedlaya, Paul Zeitz .
• Mathematical Olympiad Treasures - Titu Andreescu, Bogdan Enescu .
• Mathematical Olympiad Challenges - Titu Andreescu, Razvan Gelca .
• Lecture Notes on Mathematical Olympiad Courses - Xu Jiagu .
• Putnam and Beyond - Titu Andreescu, Razvan Gelca .
• Hungary-Israeli Mathematics Competition - Shay Gueron .
• MAA - The Contest Problem Book (Annual High School Contests) - (four volumes) - Volumes 1, 2, 3 : Charles T. Salkind, James M. Earl , Volume 4 : Ralph A. Artino, Anthony M. Gaglione, Niel Shell .
• Mathematical Olympiad in China (2007-2008) (Problems and Solutions) - Xiong Bin, Lee Peng Yee .
• What to Solve (Problems and Suggestions For Young Mathematicians) - Judita Cofman .

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Mathematics > Analysis of PDEs

Title: solving schiffer's problem in inverse scattering theory almost surely.

Abstract: In this short note, we present a probabilistic perspective on the Schiffer's problem in the inverse scattering theory, which asks whether one can uniquely determine the shape of an unknown obstacle by a single far-field measurement. It is a longstanding problem and has received considerable studies in the literature. We show that this conjecture holds true in more general settings in the probability sense. Our new perspective has important implications from the practical viewpoint and also points an interesting direction of research for broader inverse problems.

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Introduction to Number Theory (Art of Problem Solving Introduction) Paperback – June 30, 2008

• Part of series Art of Problem Solving
• Print length 314 pages
• Language English
• Publisher Aops Inc
• Publication date June 30, 2008
• Dimensions 8.25 x 0.75 x 10.5 inches
• ISBN-10 1934124125
• ISBN-13 978-1934124123
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• Publisher ‏ : ‎ Aops Inc; 2nd edition (June 30, 2008)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 314 pages
• ISBN-10 ‏ : ‎ 1934124125
• ISBN-13 ‏ : ‎ 978-1934124123
• Item Weight ‏ : ‎ 1.1 pounds
• Dimensions ‏ : ‎ 8.25 x 0.75 x 10.5 inches
• #107 in Number Theory (Books)

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Mathew crawford.

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Chemistry Education Research and Practice

Metacognitive problem solving: exploration of students’ perspectives through the lens of multi-dimensional engagement.

Solving chemical problems entails content knowledge and mastery of problem-solving processes. However, students sometimes lack metacognitive processes required for problem solving in chemistry. This study investigated how first-year chemistry students engaged with the metacognitive problem-solving scaffold Goldilocks Help. Data was collected from an activity, which involved students reflectively comparing their problem-solving attempts to an expert solution. These comparative reflections (N = 373) were thematically analysed to investigate scaffold engagement in three dimensions: cognitive, emotional, and behavioural. Finding showed that scaffold use, coupled with self-reflection, allowed students to identify flaws in their solutions that were either problem specific or related to their problem-solving skills. Students were able to propose improvement strategies, such as posing prompting questions to themselves and finding multiple alternatives for evaluating an answer. Students, who initially lacked structured problem-solving skills, found that scaffolding helped them to slow down metacognitive processes that would otherwise be rushed through or engaged with on a surface level. Students’ resistance to the scaffold was due to fear of making a mistake or viewing the scaffold as requiring extra time and effort. Within a semester, many students demonstrated an improvement in successful and structured problem solving but some required more practice to internalise the scaffold. Our findings also indicated that students’ reflections on problem solving became more sophisticated as a result of continued exposure to the scaffold and iterative opportunities to compare their work to expert solutions, to self-assess, and to reflect. Further research on reflective writing in chemistry education should focus on the ipsative nature of such assessments, i.e. processes focussing on students’ own progress, growth, and improvement, compared to their previous performance, while recognising the power relations operationalised in course-embedded reflections. From the teaching practice perspective, having an awareness of students’ thoughts, emotions, and actions can help instructors differentiate between levels of student capabilities, mindsets, and needs for extra support, allowing teaching efforts to be directed at promoting metacognitive and structured problem solving.

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K. Vo, M. Sarkar, P. J. White and E. Yuriev, Chem. Educ. Res. Pract. , 2024, Accepted Manuscript , DOI: 10.1039/D4RP00096J

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