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32 Examples of Real-World Math Problems

• • Published: April 23, 2024
• • Last update: June 14, 2024
• • Grades: All grades

Introduction

8th grade algebra problem:

Farmer Alfred has three times as many chickens as cows. In total, there are 60 legs in the barn. How many cows does Farmer Alfred have? [1]

Does this sound like a real-world math problem to you? We’ve got chickens, cows, and Farmer Alfred – it’s a scenario straight out of everyday life, isn’t it?

But before you answer, let me ask you something: If you wanted to figure out the number of cows, would you:

• Count their legs, or
• Simply count their heads ( or even just ask Farmer Alfred, “Hey Alfred, how many cows do you have?” )

Chances are, most people would go with option B.

So, why do our math books contain so many “real-world math problems” like the one above?

In this article, we’ll dive into what truly makes a math problem a real-world challenge.

Understanding 'Problems' in Daily Life and Mathematics

The word “problem” carries different meanings in everyday life and in the realm of mathematics, which can sometimes lead to confusion. In our daily lives, when we say “ I have a problem ,” we typically mean that something undesirable has occurred – something challenging to resolve or with potential negative consequences.

For instance:

• “I have a problem because I’ve lost my wallet.”
• “I have a problem because I forgot my keys at home, and I won’t be able to get into the house when I return from school.”
• “I have a problem because I was sick and missed a few weeks of school, which means I’ll likely fail my math test.”

These are examples of everyday problems we encounter. However, once the problem is solved, it often ceases to be a problem:

• “I don’t have a problem getting into my house anymore because my mom gave me her keys.”

In mathematics, the term “problem” takes on a different meaning. According to the Cambridge dictionary [2], it’s defined as “ a question in mathematics that needs an answer. “

Here are a few examples:

• If x + 2 = 4, what is the value of x?
• How do you find the common denominator for fractions 1/3 and 1/4?
• What is the length of the hypotenuse in a right triangle if two legs are 3 and 4 feet long?

These are all examples of math problems.

It’s important to note that in mathematics, a problem remains a problem even after it’s solved. Math problems are universal, regardless of who encounters them [3]. For instance, both John and Emma could face the same math problem, such as “ If x + 2 = 4, then what is x? ” After they solve it, it still remains a math problem that a teacher could give to somebody else.

Understanding Real-World Math Problems

So, what exactly is a “real-world math problem”? We’ve established that in our daily lives, we refer to a situation as a “problem” when it could lead to unpleasant consequences. In mathematics, a “problem” refers to a mathematical question that requires a mathematical solution.

With that in mind, we can define a real-world math problem as:

A situation that could have negative consequences in real life and that requires a mathematical solution (i.e. mathematical solution is preferred over other solutions).

Consider this example:

Could miscalculating the flour amount lead to less-than-ideal results? Absolutely. Messing up the flour proportion could result in a less tasty cake – certainly an unpleasant consequence.

Now, let’s explore the methods for solving this. While traditional methods, like visually dividing the flour or measuring multiple times, have their place, they may not be suitable for larger-scale events, such as catering for a wedding with 250 guests. In such cases, a mathematical solution is not only preferred but also more practical.

By setting up a simple proportion – 1 1/2 cups of flour for 8 people equals x cups for 20 people – we can quickly find the precise amount needed: 3 3/4 cups of flour. In larger events, like the wedding, the mathematical approach provides even more value, yielding a requirement of nearly 47 cups of flour.

This illustrates why a mathematical solution is faster, more accurate, and less error-prone, making it the preferred method. Coupled with the potential negative consequences of inaccuracies, this makes the problem a real-world one, showcasing the practical application of math in everyday scenarios.

Identifying Non-Real-World Math Problems

Let’s revisit the example from the beginning of this article:

Farmer Alfred has three times as many chickens as cows. In total, there are 60 legs in the barn. How many cows does Farmer Alfred have?

3rd grade basic arithmetic problem:

Noah has $56, and Olivia has 8 times less. How much money does Olivia have?

Once more, in our daily routines, do we handle money calculations this way? Why would one prefer to use mathematical solution ($56 : 8 =$7). More often than not, we’d simply ask Olivia how much money she has. While the problem can theoretically be solved mathematically, it’s much more practical, efficient, and reliable to resolve it by directly asking Olivia.

Examples like these are frequently found in math textbooks because they aid in developing mathematical thinking. However, the scenarios they describe are uncommon in real life and fail to explicitly demonstrate the usefulness of math. In essence, they don’t showcase what math can actually be used for. Therefore, there are two crucial aspects of true real-world math problems: they must be commonplace in real life, and they must explicitly illustrate the utility of math.

To summarize:

Word problems that are uncommon in real life and fail to convincingly demonstrate the usefulness of math are not real-world math problems.

Real-World Math Problems Across Professions

Many professions entail encountering real-world math problems on a regular basis. Consider the following examples:

Nurses often need to calculate accurate drug dosage amounts using proportions, a task solved through mathematical methods. Incorrect dosage calculations can pose serious risks to patient health, leading to potentially harmful consequences.

Construction engineers frequently need to determine whether a foundation will be sturdy enough to support a building, employing mathematical solving techniques and specialized formulas. Errors in these calculations can result in structural issues such as cracks in walls due to foundation deformation, leading to undesirable outcomes.

Marketers often rely on statistical analysis to assess the performance of online advertisements, including metrics like click-through rates and the geographical distribution of website visitors. Mathematical analysis guides decision-making in this area. Inaccurate analyses may lead to inefficient allocation of advertisement budgets, resulting in less-than-optimal outcomes.

These examples illustrate real-world math problems encountered in various professions. While there are numerous instances of such problems, they are often overlooked in educational settings. At DARTEF, we aim to bridge this gap by compiling a comprehensive list of real-world math problems, which we’ll explore in the following section.

32 Genuine Real-World Math Problems

In this section, we present 32 authentic real-world math problems from diverse fields such as safety and security, microbiology, architecture, engineering, nanotechnology, archaeology, creativity, and more. Each of these problems meets the criteria we’ve outlined previously. Specifically, a problem can be classified as a real-world math problem if:

• It is commonly encountered in real-life scenarios.
• It has the potential for undesirable consequences.
• A mathematical solution is preferable over alternative methods.
• It effectively illustrates the practical utility of mathematics.

All these problems stem from actual on-the-job situations, showcasing the application of middle and high school math in various professions.

Math Problems in Biology

Real-world math problems in biology often involve performing measurements or making predictions. Mathematics helps us understand various biological phenomena, such as the growth of bacterial populations, the spread of diseases, and even the reconstruction of ancient human appearances. Here are a couple of specific examples:

1. Reconstructing Human Faces Using Parallel and Perpendicular Lines:

Archeologists and forensic specialists often reconstruct human faces based on skeletal remains. They utilize parallel and perpendicular lines to create symmetry lines on the face, aiding in the recovery of facial features and proportions. Our article, “ Parallel and Perpendicular Lines: A Real-Life Example (From Forensics and Archaeology) ,” provides a comprehensive explanation and illustrates how the shape of the nose can be reconstructed using parallel and perpendicular lines, line segments, tangent lines, and symmetry lines.

2. Measuring Bacteria Size Using Circumference and Area Formulas:

Microbiology specialists routinely measure bacteria size to monitor and document their growth rates. Since bacterial shapes often resemble geometric shapes studied in school, mathematical methods such as calculating circumference or area of a circle are convenient for measuring bacteria size. Our article, “ Circumference: A Real-Life Example (from Microbiology) ,” delves into this process in detail.

Math Problems in Construction and Architecture

Real-world math problems are abundant in architecture and construction projects, where mathematics plays a crucial role in ensuring safety, efficiency, and sustainability. Here are some specific case studies that illustrate the application of math in these domains:

3. Calculating Central Angles for Safe Roadways:​

Central angle calculations are a fundamental aspect of roadway engineering, particularly in designing curved roads. Civil engineers use these calculations to determine the degree of road curvature, which significantly impacts road safety and compliance with regulations. Mathematical concepts such as radius, degrees, arc length, and proportions are commonly employed by civil engineers in their daily tasks. Our article, “ How to Find a Central Angle: A Real-Life Example (from Civil Engineering) ,” provides further explanation and demonstrates the calculation of central angles using a real road segment as an example.

4. Designing Efficient Roof Overhang Using Trigonometry

Trigonometry is a powerful tool in architecture and construction, providing a simple yet effective way to calculate the sizes of building elements, including roof overhangs. For example, the tangent function is used to design the optimal length of a roof overhang that blocks the high summer sun while allowing the lower winter sun to enter . Our article, “ Tangent (Trigonometry): A Real-Life Example (From Architecture and Construction) ,” provides a detailed use case with two methods of calculation and also includes a worksheet.

5. Designing Efficient Roofs for Solar Panels Using Angle Geometry:

Mathematics plays a crucial role in architecture, aiding architects and construction engineers in designing energy-efficient building structures. For example, when considering the installation of solar panels on a building’s roof, understanding geometric properties such as adjacent and alternate angles is essential for maximizing energy efficiency . By utilizing mathematical calculations, architects can determine the optimal angle for positioning solar panels and reflectors to capture maximum sunlight. Our article, “ A Real-Life Example of How Angles are Used in Architecture ,” provides explanations and illustrations, including animations.

6. Precision Drilling of Oil Wells Using Trigonometry:

Petroleum engineers rely on trigonometric principles such as sines, cosines, tangents, and right-angle triangles to drill oil wells accurately. Trigonometry enables engineers to calculate precise angles and distances necessary for drilling vertical, inclined, and even horizontal wells. For instance, when drilling at an inclination of 30°, engineers can use trigonometry to calculate the vertical depth corresponding to each foot drilled horizontally. Our article, “ CosX: A Real-Life Example (from Petroleum Engineering) ,” provides examples, drawings of right triangle models, and necessary calculations.

7. Calculating Water Flow Rate Using Composite Figures:

Water supply specialists frequently encounter the task of calculating water flow rates in water canals, which involves determining the area of composite figures representing canal cross-sections . Many canal cross-sections consist of composite shapes, such as rectangles and triangles. By calculating the areas of these individual components and summing them, specialists can determine the total cross-sectional area and subsequently calculate the flow rate of water. Our article, “ Area of Composite Figures: A Real-Life Example (from Water Supply) ,” presents necessary figures, cross-sections, and an example calculation.

Math Problems in Business and Marketing

Mathematics plays a crucial role not only in finance and banking but also in making informed decisions across various aspects of business development, marketing analysis, growth strategies, and more. Here are some real-world math problems commonly encountered in business and marketing:

8. Analyzing Webpage Position in Google Using Polynomials and Polynomial Graphs:

Polynomials and polynomial graphs are essential tools for data analysis. They are useful for a wide variety of people, not only data analysts, but literally everyone who ever uses Excel or Google Sheets. Our article, “ Graphing Polynomials: A Real-World Example (from Data Analysis) ,” describes this and provides an authentic case study. The case study demonstrates how polynomials, polynomial functions of varying degrees, and polynomial graphs are used in internet technologies to analyze the visibility of webpages in Google and other search engines.”

9. Making Informed Business Development Decisions Using Percentages:

In business development, math problems often revolve around analyzing growth and making strategic decisions. Understanding percentages is essential, particularly when launching new products or services. For example, determining whether a startup should target desktop computer, smartphone, or tablet users for an app requires analyzing installation percentages among user groups to gain insights into consumer behavior. Math helps optimize marketing efforts, enhance customer engagement, and drive growth in competitive markets. Check out our article “ A Real-Life Example of Percent Problems in Business ” for a detailed description of this example.

10. Analyzing Customer Preferences Using Polynomials:

Marketing involves not only creative advertising but also thorough analysis of customer preferences and marketing campaigns. Marketing specialists often use simple polynomials for such analysis, as they help analyze multiple aspects of customer behavior simultaneously. For instance, marketers may use polynomials to determine whether low price or service quality is more important to hotel visitors. Smart analysis using polynomials enables businesses to make informed decisions. If you’re interested in learning more, our article “ Polynomials: A Real Life Example (from Marketing) ” provides a real-world example from the hotel industry.

11. Avoiding Statistical Mistakes Using Simpson’s Paradox:

Data gathering and trend analysis are essential in marketing, but a good understanding of mathematical statistics helps avoid intuitive mistakes . For example, consider an advertising campaign targeting Android and iOS users. Initial data may suggest that iOS users are more responsive. However, a careful statistical analysis, as described in our article “ Simpson’s Paradox: A Real-Life Example (from Marketing) ,” may reveal that Android users, particularly tablet users, actually click on ads more frequently. This contradiction highlights the importance of accurate statistical interpretation and the careful use of mathematics in decision-making.

Math Problems in Digital Agriculture

In the modern era, agriculture is becoming increasingly digitalized, with sensors and artificial intelligence playing vital roles in farm management. Mathematics is integral to this digital transformation, aiding in data analysis, weather prediction, soil parameter measurement, irrigation scheduling, and more. Here’s an example of a real-world math problem in digital agriculture:

12. Combatting Pests with Negative Numbers:

Negative numbers are utilized in agriculture to determine the direction of movement, similar to how they’re used on a temperature scale where a plus sign indicates an increase and a minus sign indicates a decrease. In agricultural sensors, plus and minus signs may indicate whether pests are moving towards or away from plants. For instance, a plus sign could signify movement towards plants, while a minus sign indicates movement away from plants. Understanding these directional movements helps farmers combat pests effectively. Our article, “ Negative Numbers: A Real-Life Example (from Agriculture) ,” provides detailed explanations and examples of how negative numbers are applied in agriculture.

Math Problems in DIY Projects

In do-it-yourself (DIY) projects, mathematics plays a crucial role in measurements, calculations, and problem-solving. Whether you’re designing furniture, planning home renovations, crafting handmade gifts, or landscaping your garden, math provides essential tools for precise measurements, material estimations, and budget management. Here’s an example of a real-world math problem in DIY:

13. Checking Construction Parts for Right Angles Using the Pythagorean Theorem:

The converse of the Pythagorean theorem allows you to check whether various elements – such as foundations, corners of rooms, garage walls, or sides of vegetable patches – form right angles. This can be done quickly using Pythagorean triples like 3-4-5 or 6-8-10, or by calculating with square roots. This method ensures the creation of right angles or verifies if an angle is indeed right. Our article, “ Pythagorean Theorem Converse: A Real-Life Example (from DIY) ,” explains this concept and provides an example of how to build a perfect 90° foundation using the converse of the Pythagorean theorem.

Math Problems in Entertainment and Creativity

Mathematics plays a surprisingly significant role in the creative industries, including music composition, visual effects creation, and computer graphic design. Understanding and applying mathematical concepts is essential for producing engaging and attractive creative works. Here’s an example of a real-world math problem in entertainment:

14. Controlling Stage Lamps with Linear Functions:

Linear and quadratic functions are essential components of the daily work of lighting specialists in theaters and event productions. These professionals utilize specialized software and controllers that rely on algorithms based on mathematical functions. Linear functions, in particular, are commonly used to control stage lamps, ensuring precise and coordinated lighting effects during performances. Our article, “ Linear Function: A Real-Life Example (from Entertainment) ,” delves into this topic in detail, complete with animations that illustrate how these functions are applied in practice.

Math Problems in Healthcare

Mathematics plays a crucial role in solving numerous real-world problems in healthcare, ranging from patient care to the design of advanced medical devices. Here are several examples of real-world math problems in healthcare:

15. Calculating Dosage by Converting Time Units:

Nurses frequently convert between hours, minutes, and seconds to accurately administer medications and fluids to patients. For example, if a patient requires 300mL of fluid over 2.5 hours, nurses must convert hours to minutes to calculate the appropriate drip rate for an IV bag. Understanding mathematical conversions and solving proportions are essential skills in nursing. Check out our article, “ Converting Hours to Minutes: A Real-Life Example (from Nursing) ,” for further explanation.

16. Predicting Healthcare Needs Using Quadratic Functions:

Mathematical functions, including quadratic functions, are used to make predictions in public health. These predictions are vital for estimating the need for medical services, such as psychological support following traumatic events. Quadratic functions can model trends in stress symptoms over time, enabling healthcare systems to anticipate and prepare for increased demand. Our article, “ Parabola Equation: A Real-Life Example (from Public Health) ,” provides further insights and examples.

17. Predicting Healthcare Needs Using Piecewise Linear Functions:

Piecewise linear functions are useful for describing real-world trends that cannot be accurately represented by other functions. For instance, if stress symptoms fluctuate irregularly over time, piecewise linear functions can define periods of increased and decreased stress levels. Our article, “ Piecewise Linear Function: A Real-Life Example (From Public Health) ,” offers an example of how these functions are applied.

18. Designing Medical Prostheses Using the Pythagorean Theorem:

The Pythagorean theorem is applied in the design of medical devices , particularly prostheses for traumatic recovery. Components of these devices often resemble right triangles, allowing engineers to calculate movement and placement for optimal patient comfort and stability during rehabilitation. For a detailed explanation and animations, see our article, “ The 3-4-5 Triangle: A Real-Life Example (from Mechatronics) .”

19. Illustrating Disease Survival Rates Using Cartesian Coordinate Plane:

In medicine, Cartesian coordinate planes are utilized for analyzing historical data, statistics, and predictions. They help visualize relationships between independent and dependent variables, such as survival rates and diagnostic timing in diseases like lung cancer. Our article, “ Coordinate Plane: A Real-Life Example (from Medicine) ,” offers a detailed exploration of this concept, including analyses for both non-smoking and smoking patients.

Math Problems in Industry

Mathematics plays a vital role in solving numerous real-world problems across various industries. From designing industrial robots to analyzing production quality and planning logistics, math is indispensable for optimizing processes and improving efficiency. Here are several examples of real-world math problems in industry:

20. Precise Movements of Industrial Robots Using Trigonometry:

Trigonometry is extensively utilized to direct and control the movements of industrial robots. Each section of an industrial robot can be likened to a leg or hypotenuse of a right triangle, allowing trigonometry to precisely control the position of the robot head. For instance, if the robot head needs to move 5 inches to the right, trigonometry enables calculations to ensure each section of the robot moves appropriately to achieve the desired position. For detailed examples and calculations, refer to our article, “ Find the Missing Side: A Real-life Example (from Robotics) .”

21. Measuring Nanoparticle Size via Cube Root Calculation:

Mathematics plays a crucial role in nanotechnology, particularly in measuring the size of nanoparticles. Cube roots are commonly employed to calculate the size of cube-shaped nanoparticles. As nanoparticles are extremely small and challenging to measure directly, methods in nanotechnology determine nanoparticle volume, allowing for the calculation of cube root to determine the size of the nanoparticle cube side. Check out our article, “ Cube Root: A Real-Life Example (from Nanotechnology) ,” for explanations, calculations, and insights into the connection between physics and math.

22. Understanding Human Emotions Using Number Codes for Robots:

Mathematics is employed to measure human emotions effectively, which is essential for building smart robots capable of recognizing and responding to human emotions. Each human emotion can be broken down into smaller features, such as facial expressions, which are assigned mathematical codes to create algorithms for robots to recognize and distinguish emotions. Explore our article, “ Psychology and Math: A Real-Life Example (from Smart Robots) ,” to understand this process and its connection to psychology.

23. Systems of Linear Equations in Self-Driving Cars:

In the automotive industry, mathematics is pivotal for ensuring the safety and efficiency of self-driving cars. Systems of linear equations are used to predict critical moments in road safety, such as when two cars are side by side during an overtaking maneuver. By solving systems of linear equations, self-driving car computers can assess the safety of overtaking situations. Our article, “ Systems of Linear Equations: A Real-Life Example (from Self-Driving Cars) ,” provides a comprehensive explanation, including animated illustrations and interactive simulations, demonstrating how linear functions and equations are synchronized with car motion.

Math Problems in Information Technology (IT)

Mathematics serves as a fundamental tool in the field of information technology (IT), underpinning various aspects of software development, cybersecurity, and technological advancements. Here are some real-world math problems encountered in IT:

24. Selecting the Right Word in Machine Translation Using Mathematical Probability:

Theoretical probability plays a crucial role in AI machine translation systems, such as Google Translate, by aiding in the selection of the most appropriate words. Words often have multiple translations, and theoretical probability helps analyze the frequency of word appearances in texts to propose the most probable translation. Dive deeper into this topic in our article, “ Theoretical Probability: A Real-Life Example (from Artificial Intelligence) .”

25. Creating User-Friendly Music Streaming Websites Using Mathematical Probability:

Probability is utilized in user experience (UX) design to enhance the usability of digital products , including music streaming websites. Recommendation algorithms calculate theoretical probability to suggest songs that users are likely to enjoy, improving their overall experience. Learn more about this application in our article, “ Theoretical Probability: A Real-Life Example (from Digital Design) .”

26. Developing Realistic Computer Games with Vectors:

Mathematics is essential in animating objects and simulating real-world factors in computer games. Vectors enable game developers to incorporate realistic elements like gravity and wind into the gaming environment. By applying vector addition, developers can accurately depict how external forces affect object trajectories, enhancing the gaming experience. Explore this concept further in our article, “ Parallelogram Law of Vector Addition: A Real-Life Example (from Game Development) .”

27. Detecting Malicious Bots in Social Media Using Linear Inequalities:

Mathematical inequalities are valuable tools in cybersecurity for identifying malicious activity on social media platforms. By analyzing behavioral differences between real users and bots—such as friend count, posting frequency, and device usage—cybersecurity experts can develop algorithms to detect and block suspicious accounts. Discover more about this application in our article, “ How to Write Inequalities: A Real-life Example (from Social Media) .”

28. Increasing Computational Efficiency Using Algebraic and Rational Expressions

Algebraic and rational expressions are crucial in computer technology for increasing computational efficiency . Every step in a computer program’s algorithm requires valuable time and energy for execution. This is especially important for programs used in various safety and security systems. Simplifying these expressions helps boost computational performance. Our article, “ Algebraic and Rational Expressions: A Real-Life Example (from Computer Technology) ,” explains this in detail.

Math Problems in Legal Issues

In legal proceedings, mathematics plays a crucial role in analyzing data and making informed decisions. Here’s a real-world math problem encountered in legal work:

29. Proving the Reliability of Technology in Court Using Percentages:

In certain court cases, particularly those involving new technologies like autonomous cars, lawyers may need to employ mathematical methods to justify evidence. Percentage analysis can be utilized to assess the reliability of technology in court. For instance, in cases related to self-driving cars, lawyers may compare the percentage of errors made by autonomous vehicles with those made by human drivers to determine the technology’s safety. Delve deeper into this topic in our article, “ Solving Percent Problems: A Real-Life Example (from Legal) .”

Math Problems in Safety and Security

In the realm of safety and security, mathematics plays a vital role in protecting people, nature, and assets. Real-world math problems in this field can involve reconstructing crime scenes, analyzing evidence, creating effective emergency response plans, and predicting and responding to natural disasters. Here are two examples:

30. Reconstructing a Crime Scene with Inverse Trigonometry:

Forensic specialists rely on mathematical methods, such as inverse trigonometry, to uncover details of crimes long after they’ve occurred. Inverse trigonometric functions like arcsin, arccos, and arctan enable forensic specialists to calculate precise shooting angles or the trajectory of blood drops at crime scenes. Dive into this topic in our article, “ Arctan: A Real-Life Example (from Criminology) .”

31. Responding to Wildfires with Mathematical Variables:

Variables in mathematical language are also prevalent in real-world scenarios like firefighting. For instance, when dealing with wildfires, independent variables like forest type, wind speed and direction, rainfall, and terrain slope influence fire spread speed (dependent variable) . By utilizing specialized formulas incorporating these variables, fire protection specialists can accurately predict fire paths and optimize firefighting efforts. Explore this practical application in our article, “ Dependent and Independent Variables: A Real-Life Example (from Fire Protection) .”

Math Problems in Weather and Climate

In the realm of weather and climate, mathematics is crucial for creating mathematical models of Earth’s atmosphere, making weather forecasts, predicting weather patterns, and assessing long-term climate trends. Here’s an example:

32. Forecasting Thunderstorms with Negative Numbers:

Negative numbers play a significant role in forecasting thunderstorms. The probability of thunderstorms is closely tied to the temperature differences between air masses in the atmosphere. By subtracting negative numbers (representing temperature values) and analyzing the results, meteorologists can forecast the likelihood of thunderstorms based on the temperature differentials. Dive deeper into this concept in our article, “ Subtracting Negative Numbers: A Real-Life Example (from Weather Forecast). “

Conclusions

Many students perceive math as disconnected from the real world, leading them to question its relevance. Admittedly, maintaining this relevance is no easy feat—it requires collective effort. Farmer Alfred can’t tackle this challenge alone. However, introducing more real-world math examples into the school curriculum is a crucial step. Numerous studies have demonstrated that such examples significantly enhance students’ motivation to learn math. Our own pilot tests confirm this, showing that a clear connection between real-world problems and math education profoundly influences how young people view their future careers.

Therefore, to increase the overall relevance of math education and motivate students, we must prioritize the integration of real-world examples into our teaching practices. At DARTEF, we are dedicated to this cause, striving to bring authenticity to mathematics education worldwide.

• Vos, Pauline. ““How real people really need mathematics in the real world”—Authenticity in mathematics education.” Education Sciences  8.4 (2018): 195.
• Cambridge dictionary, Meaning of problem in English.
• Reif, Frederick.  Applying cognitive science to education: Thinking and learning in scientific and other complex domains . MIT press, 2008.

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About Getting Smart

Getting smart collective, impact update, 7 real-world issues that can allow students to tackle big challenges.

Ever since I started teaching in 1990, I have been a student voice advocate. Whether it was as a media/English teacher, student leadership advisor or a site leader. I have always believed that students not only have good ideas, but that they may just have new, unique or even better ones. In an effort to find their own voice and place in the world, they may see things that we don’t see or have long been paralyzed to do anything about. In 1999, I saw students address a school’s racial divide and cultural issues by creating a school-wide learning experience (see Harmony at Buchanan High School ). Ever since then, I have believed that projects with real-world outcomes hold some of the greatest potential for helping students become driven, empathetic and engaged citizens. The outpouring of student voice  in the wake of the recent tragedy in Parkland, Florida, is a great example.

When we begin the project design process in PBL, we can start either with a challenging problem or question and then tie it to our standards, or we can start with our standards and connect them to a real-world challenge. This second approach is more foundational to project based learning, for many reasons, including student engagement, student voice, relevance and authenticity. But beyond that, we also do it because this is where jobs are. Jobs are created and grown as we work to address the real problems facing our world and peoples. Our students are ready to tackle the problems facing our world. They have a voice. They have the tools and resources. And they are not afraid to collaborate and form new communities poised for the problem-solving work that needs to be done.

As an educator, parent and advocate for an engaged/empowered citizenry, I could not be prouder of how the students in Parkland, Florida – along with their peers across the nation – have both found their voice, as well as changed the narrative. These students, as well as many others across the nation, are not afraid to collaborate, and use new technologies and form new professional networks in order to address our current and future challenges. Let’s be honest, our best hope of improving the status of our planet’s many issues truly lie with our youth.

With all of this in mind, there are a number of current and ongoing real-world challenges that we currently face (and probably will for a long time). I don’t like the term “problem-solving” in this context, as it implies that we can fix, cure or eradicate a problem or challenge, but by going after our problems with new solutions, we can certainly move progress forward. And in that movement, there is magic. There is innovation. There is change. There is our collective human mission: how can we creatively collaborate, critically think and communicate in ways that make our world a better place to live.

New Pathways Handbook

Over the last few years, we’ve shared hundreds of stories about connecting students to work (and skills) that matters through our blog, podcast, and various publications. To synthesize these key learnings, we compiled the New Pathways Handbook, a great jumping-off point to our numerous resources and launchpad for getting started with pathways.

Our students are ready to exercise their collective voices and create calls to action. The following seven ideas are not ranked, but are rather my go to “top seven” that naturally lend themselves to projects that excite student interest, rely on available resources, and maintain relevance and authenticity. Moreover, they are not subject-specific. Indeed, there are many opportunities for English, science, social science, math and others to connect to these project challenges. They are:

1) Climate Change – Climate Change will have a significant impact on our students’ lives. Indeed, there may not be one issue that will impact them more comprehensively. Students have seen the data and witnessed the changes, and are listening to the science community. They know that this an urgent issue that will affect almost everything, including, but not limited to, weather, sea levels, food security, water quality, air quality, sustainability and much more. Many organizations – such as NASA , The National Park Service , National Center for Science Education , National Oceanic Atmospheric Association  and SOCAN  to name a few – are working to bring climate change curriculum and projects to teachers and students.

2) Health Care  – Since this has become a prominent topic in the national debate, students are becoming aware of the issues in our country related to rising costs, access, quality and equity. They are beginning to understand the importance both individually and societally. Like the aforementioned topic of climate change, students are also (and unfortunately) learning that we are not necessarily leading the world in this area. They know that this problem is connected to profits, insurance, bureaucracy and more, but they also have a fresher sense of how it could be different, and how we could learn from others around the world. The work on this topic, like many others, is being led by our universities. Institutions such as University of Michigan , Johns Hopkins  and Stanford are leading the way.

3) Food Insecurity   – as our students become more aware of their surrounding communities, as well as the peers they interact with daily, they begin to see differences. Differences in socioeconomic status, opportunities for growth, housing, security, support services and more. And since 13 million young people live in food-insecure homes, almost all of our students, as well as educators, know someone who is hungry on a daily basis. This may often start with service-based projects, but can also lead to high quality project based learning complete with research, data analysis, diverse solutions and ultimately a variety of calls to action. If you want to see how one teacher and his students transformed not only their school, but entire community related to food insecurity, check out Power Of A Plant author Stephen Ritz and the Green Bronx Machine .

4) Violence  – This is a natural given current events taking the nation by storm. However, the related topics and issues here are not new. And yes, they are politically charged, but young people care about these issues . They care about their collective safety and futures, but also know something can be done. In addition to the specifics related to school violence and safety, students can study details of how to advocate, organize, campaign and solicit support, learn that this is a complex problem that has many plausible causes, and, perhaps most importantly, hope for progress. They also know that although they are concerned about attending school in safe environments, our society and culture have violence-related problems and issues that they want to see addressed. Following the recent incident in Florida and the subsequent response from students, the New York Times has compiled a list of resources  for educators on this topic.

5) Homelessness  – We often hear the expression “think globally, act locally.” The topic of homelessness has garnered more attention than ever as more and more communities wrestle with a growing homeless population. In addition to opportunities for our students and schools to partner with local non-profit organizations dealing with homelessness, this topic, like others, is also a great way to elicit empathy in our students. We often hear from educators, employers and others that we want to raise adults that are able to solve problems, improve our communities, and have the ability to see beyond themselves. This topic can provide a number of options for helping students develop those skills. Finally, we also have a growing population of homeless students. So, the relevancy and urgency are all there. Many have laid the groundwork for us to address this within our curriculum. Organizations like Bridge Communities , National Coalition For The Homeless , Homeless Hub  and Learning To Give  are some of the many leading the way.

6) Sustainability  – This is an extremely global issue that affects everything from energy, to food, to resources, economics, health, wellness and more. Students are becoming more and more aware that our very future as a species depends on how we address sustainability challenges. They are aware that this challenge requires new ways of thinking, new priorities, new standards and new ways of doing things. Sustainability is all about future innovation. Students have tremendous opportunities to collaborate, think critically, communicate, and be creative when questioning if a current practice, method, resource or even industry is sustainable without dramatic change and shifts. Students who tackle these challenges will be our leaders – business, political and cultural – of the future. Educators and students can find almost infinite resources and partners. A few of these are Green Education Foundation , Green Schools Initiative , Strategic Energy Innovations , Facing the Future  and Teach For America .

7) Education  – It seems that each and every day, more and more of us (though maybe still not enough) are moving closer to realizing that our educational systems are seemingly unprepared to make the big shifts needed to truly address the learning needs of 21st-century students. The related challenges are many – new literacies, skills, economic demands, brain research, technology, outcomes and methodologies. It’s a good thing that more and more people – both inside and outside of education – are both demanding and implementing change. However, one of the continued ironies within education is that we (and I recognize that this is a generalization) rarely ask the primary customer (students) what they think their education should look, feel and sound like. We have traditionally underestimated their ability to articulate what they need and what would benefit them for their individual and collective futures. One of the many foundational advantages of project based learning is that we consult and consider the student in project design and implementation. Student “voice & choice” creates opportunities for students to have input on and make decisions regarding everything from the final product, to focus area within a topic or challenge, and even whom they may partner with from peers to professionals. It’s this choice that not only helps elicit engagement and ownership of learning, but offers opportunities for students to enhance all of the skills that we want in our ideal graduates. As one might guess, there is not a lot of formal curriculum being developed for teachers to lead students through the issue of education reform. This may need to be an organic thing that happens class by class and school by school. It can start as easily as one teacher asking students about what they want out of their education. Some other entry points are The Buck Institute for Education , Edutopia’s Five Ways To Give Your Students More Voice & Choice , Barbara Bray’s Rethinking Learning  and reDesign .

This is not intended to be an exhaustive or comprehensive list. However, these seven broad topics present hundreds of relevant challenges that our students can and should have opportunities to address. If they do, they will not only be more prepared for their futures, but also poised to positively impact all of our futures.

For more, see:

• High Quality PBL Case Study: School21
• In Broward County, Student Voice Impacts the Classroom and Beyond
• Introducing a Framework for High Quality Project Based Learning

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What is pbe (spanish), designing microschools download, download quick start guide to implementing place-based education, download quick start guide to place-based professional learning, download what is place-based education and why does it matter, download 20 invention opportunities in learning & development.

ChatGPT for Teachers

Trauma-informed practices in schools, teacher well-being, cultivating diversity, equity, & inclusion, integrating technology in the classroom, social-emotional development, covid-19 resources, invest in resilience: summer toolkit, civics & resilience, all toolkits, degree programs, trauma-informed professional development, teacher licensure & certification, how to become - career information, classroom management, instructional design, lifestyle & self-care, online higher ed teaching, current events, stem projects that tackle real-world problems.

STEM learning is largely about designing creative solutions for real-world problems. When students learn within the context of authentic, problem-based STEM design, they can more clearly see the genuine impact of their learning. That kind of authenticity builds engagement, taking students from groans of “When will I ever use this?” to a genuine connection between skills and application.

Using STEM to promote critical thinking and innovation

“Educational outcomes in traditional settings focus on how many answers a student knows. We want students to learn how to develop a critical stance with their work: inquiring, editing, thinking flexibly, and learning from another person’s perspective,” says Arthur L. Costa in his book Learning and Leading with Habits of Mind . “The critical attribute of intelligent human beings is not only having information but also knowing how to act on it.”

Invention and problem-solving aren’t just for laboratory thinkers hunkered down away from the classroom. Students from elementary to high school can wonder, design, and invent a real product that solves real problems. “ Problem-solving involves finding answers to questions and solutions for undesired effects. STEM lessons revolve around the engineering design process (EDP) — an organized, open-ended approach to investigation that promotes creativity, invention, and prototype design, along with testing and analysis,” says Ann Jolly in her book STEM by Design . “These iterative steps will involve your students in asking critical questions about the problem, and guide them through creating and testing actual prototypes to solve that problem.”

STEM projects that use real-world problems

Here are some engaging projects that get your students thinking about how to solve real-world problems.

Preventing soil erosion

In this project, meant for sixth – 12th grade, students learn to build a seawall to protest a coastline from erosion, calculating wave energy to determine the best materials for the job.  See the project.

Growing food during a flood

A natural disaster that often devastates communities, floods can make it difficult to grow food. In this project, students explore “a problem faced by farmers in Bangladesh and how to grow food even when the land floods.”  See the project .

Solving a city’s design needs

Get your middle or high school students involved in some urban planning. Students can identify a city’s issues, relating to things like transportation, the environment, or overcrowding — and design solutions. See the project here or this Lego version for younger learners.

Creating clean water

Too many areas of the world — including cities in our own country — do not have access to clean water. In this STEM project, teens will learn how to build and test their own water filtration systems.  See the project here .

Improving the lives of those with disabilities

How can someone with crutches or a wheelchair carry what they need? Through some crafty designs! This project encourages middle school students to think creatively  and  to participate in civic engagement.   See the project here .

Cleaning up an oil spill

We’ve all seen images of beaches and wildlife covered in oil after a disastrous spill. This project gets elementary to middle school students designing and testing oil spill clean-up kits. See the project here .

Building earthquake-resistant structures

With the ever-increasing amount of devastating earthquakes around the world, this project solves some major problems. Elementary students can learn to create earthquake resistant structures in their classroom. See the project here .

Constructing solar ovens

In remote places or impoverished areas, it’s possible to make solar ovens to safely cook food. In this project, elementary students construct solar ovens to learn all about how they work and their environmental and societal impact.  See the project here .

Stopping apple oxidization

Stop those apples from turning brown with this oxidation-based project. Perfect for younger learners, students can predict, label, count, and experiment! See the project here .

Advancing as a STEAM educator

The push for STEM has evolved into the STEAM movement, adding the arts for further enrichment and engagement. There are so many ways to embed STEM or STEAM lessons in your curriculum, but doing it well requires foundational knowledge and professional development. Imagine what type of impact you could have on your students and your community if you were supported by a theoretical framework, a variety of strategies, and a wealth of ideas and resources.

You may also like to read

• Teaching STEM: Challenging Students to Think Through Tough Problems
• Professional Development Resources for STEM Teachers
• What is the Washington State STEM Lighthouse Program?
• Characteristics of a Great STEAM Program
• Building a Partnership Between Your School and a STEAM Organization
• The Art of Inquiry in STEAM Education

Categorized as: Tips for Teachers and Classroom Resources

Tagged as: Art ,  Educational Technology ,  Engaging Activities ,  Math and Science ,  Science ,  STEAM

• Online & Campus Master's in Elementary Educat...
• 2020 Civics Engagement & Resilience: Tools fo...
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5 Effective Problem-Solving Strategies

Got a problem you’re trying to solve? Strategies like trial and error, gut instincts, and “working backward” can help. We look at some examples and how to use them.

We all face problems daily. Some are simple, like deciding what to eat for dinner. Others are more complex, like resolving a conflict with a loved one or figuring out how to overcome barriers to your goals.

No matter what problem you’re facing, these five problem-solving strategies can help you develop an effective solution.

What are problem-solving strategies?

To effectively solve a problem, you need a problem-solving strategy .

If you’ve had to make a hard decision before then you know that simply ruminating on the problem isn’t likely to get you anywhere. You need an effective strategy — or a plan of action — to find a solution.

In general, effective problem-solving strategies include the following steps:

• Define the problem.
• Come up with alternative solutions.
• Decide on a solution.
• Implement the solution.

Problem-solving strategies don’t guarantee a solution, but they do help guide you through the process of finding a resolution.

Using problem-solving strategies also has other benefits . For example, having a strategy you can turn to can help you overcome anxiety and distress when you’re first faced with a problem or difficult decision.

The key is to find a problem-solving strategy that works for your specific situation, as well as your personality. One strategy may work well for one type of problem but not another. In addition, some people may prefer certain strategies over others; for example, creative people may prefer to depend on their insights than use algorithms.

It’s important to be equipped with several problem-solving strategies so you use the one that’s most effective for your current situation.

1. Trial and error

One of the most common problem-solving strategies is trial and error. In other words, you try different solutions until you find one that works.

For example, say the problem is that your Wi-Fi isn’t working. You might try different things until it starts working again, like restarting your modem or your devices until you find or resolve the problem. When one solution isn’t successful, you try another until you find what works.

Trial and error can also work for interpersonal problems . For example, if your child always stays up past their bedtime, you might try different solutions — a visual clock to remind them of the time, a reward system, or gentle punishments — to find a solution that works.

2. Heuristics

Sometimes, it’s more effective to solve a problem based on a formula than to try different solutions blindly.

Heuristics are problem-solving strategies or frameworks people use to quickly find an approximate solution. It may not be the optimal solution, but it’s faster than finding the perfect resolution, and it’s “good enough.”

Algorithms or equations are examples of heuristics.

An algorithm is a step-by-step problem-solving strategy based on a formula guaranteed to give you positive results. For example, you might use an algorithm to determine how much food is needed to feed people at a large party.

However, many life problems have no formulaic solution; for example, you may not be able to come up with an algorithm to solve the problem of making amends with your spouse after a fight.

3. Gut instincts (insight problem-solving)

While algorithm-based problem-solving is formulaic, insight problem-solving is the opposite.

When we use insight as a problem-solving strategy we depend on our “gut instincts” or what we know and feel about a situation to come up with a solution. People might describe insight-based solutions to problems as an “aha moment.”

For example, you might face the problem of whether or not to stay in a relationship. The solution to this problem may come as a sudden insight that you need to leave. In insight problem-solving, the cognitive processes that help you solve a problem happen outside your conscious awareness.

4. Working backward

Working backward is a problem-solving approach often taught to help students solve problems in mathematics. However, it’s useful for real-world problems as well.

Working backward is when you start with the solution and “work backward” to figure out how you got to the solution. For example, if you know you need to be at a party by 8 p.m., you might work backward to problem-solve when you must leave the house, when you need to start getting ready, and so on.

5. Means-end analysis

Means-end analysis is a problem-solving strategy that, to put it simply, helps you get from “point A” to “point B” by examining and coming up with solutions to obstacles.

When using means-end analysis you define the current state or situation (where you are now) and the intended goal. Then, you come up with solutions to get from where you are now to where you need to be.

For example, a student might be faced with the problem of how to successfully get through finals season . They haven’t started studying, but their end goal is to pass all of their finals. Using means-end analysis, the student can examine the obstacles that stand between their current state and their end goal (passing their finals).

They could see, for example, that one obstacle is that they get distracted from studying by their friends. They could devise a solution to this obstacle by putting their phone on “do not disturb” mode while studying.

Let’s recap

Whether they’re simple or complex, we’re faced with problems every day. To successfully solve these problems we need an effective strategy. There are many different problem-solving strategies to choose from.

Although problem-solving strategies don’t guarantee a solution, they can help you feel less anxious about problems and make it more likely that you come up with an answer.

8 sources collapsed

• Chu Y, et al. (2011). Human performance on insight problem-solving: A review. https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1094&context=jps
• Dumper K, et al. (n.d.) Chapter 7.3: Problem-solving in introductory psychology. https://opentext.wsu.edu/psych105/chapter/7-4-problem-solving/
• Foulds LR. (2017). The heuristic problem-solving approach. https://www.tandfonline.com/doi/abs/10.1057/jors.1983.205
• Gick ML. (1986). Problem-solving strategies. https://www.tandfonline.com/doi/abs/10.1080/00461520.1986.9653026
• Montgomery ME. (2015). Problem solving using means-end analysis. https://sites.psu.edu/psych256sp15/2015/04/19/problem-solving-using-means-end-analysis/
• Posamentier A, et al. (2015). Problem-solving strategies in mathematics. Chapter 3: Working backwards. https://www.worldscientific.com/doi/10.1142/9789814651646_0003
• Sarathy V. (2018). Real world problem-solving. https://www.frontiersin.org/articles/10.3389/fnhum.2018.00261/full
• Woods D. (2000). An evidence-based strategy for problem solving. https://www.researchgate.net/publication/245332888_An_Evidence-Based_Strategy_for_Problem_Solving

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Overview of the Problem-Solving Mental Process

• Identify the Problem
• Define the Problem
• Form a Strategy
• Organize Information
• Allocate Resources
• Monitor Progress
• Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

• Asking questions about the problem
• Breaking the problem down into smaller pieces
• Looking at the problem from different perspectives
• Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

• Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
• Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

Get Advice From The Verywell Mind Podcast

Hosted by therapist Amy Morin, LCSW, this episode of The Verywell Mind Podcast shares how you can stop dwelling in a negative mindset.

Follow Now : Apple Podcasts / Spotify / Google Podcasts

You can become a better problem solving by:

• Practicing brainstorming and coming up with multiple potential solutions to problems
• Being open-minded and considering all possible options before making a decision
• Breaking down problems into smaller, more manageable pieces
• Asking for help when needed
• Researching different problem-solving techniques and trying out new ones
• Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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Innovate learning with the 21CLD real-world problem solving and innovation dimension

This module defines real-world problem solving for educators and explains the dimensions that must be present in such classroom activities to prepare learners with 21st century skills.

Learning objectives

In this module, you will:

• Determine why real-world problem solving and innovation is important
• Define problem solving
• Explore the real-world problem solving and innovation rubric and decision tree
• Examine Microsoft tools that support real-world problem solving and innovation
• Design learning activities that focus on real-world problem solving and innovation

Prerequisites

• Introduction min
• Introduction to real-world problem solving and innovation min
• Design learning experiences with the real-world problem solving and innovation rubric min
• Support real-world problem solving and innovation with Microsoft tools min
• Real-world problem solving and innovation in action min
• Knowledge check min
• Summary min

• Grades 6-12
• School Leaders

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26 Snappy Answers to the Question “When Are We Ever Going to Use This Math in Real Life?”

Next time they ask, you’ll be ready.

As a math teacher, how many times have you heard frustrated students ask, “When are we ever going to use this math in real life!?” We know, it’s maddening! Especially for those of us who love math so much we’ve devoted our lives to sharing it with others.

It may very well be true that students won’t use some of the more abstract mathematical concepts they learn in school unless they choose to work in specific fields. But the underlying skills they develop in math class—like taking risks, thinking logically and solving problems—will last a lifetime and help them solve work-related and real-world problems.

Here are 26 images and accompanying comebacks to share with your students to get them thinking about all the different and unexpected ways they might use math in their futures!

1. If you go bungee jumping, you might want to know a thing or two about trajectories.

https://giphy.com/gifs/funny-fail-5OuUiP0we57b2

Source: GIPHY

2. When you invest your money, you’ll do better if you understand concepts such as interest rates, risk vs. reward, and probability.

3. once you’re a driver, you’ll need to be able to calculate things like reaction time and stopping distance., 4. in case of a zombie apocalypse, you’re going to want to explore geometric progressions, interpret data and make predictions in order to stay human..

Trigger an outbreak of learning and infectious fun in your classroom with this Zombie Apocalypse activity from TI’s STEM Behind Hollywood series.

5. Before you tackle that home wallpaper project, you’ll need to calculate just how much wall paper glue you need per square foot.

6. when you buy your first house and apply for a 30-year mortgage, you may be shocked by the reality of what interest compounded over 30 years looks like., 7. to be a responsible pet owner, you’ll need to calculate how much hamster food to have on hand., 8. even if you’re just an armchair athlete, you can’t believe the math involved in kicking field goals.

Check out this Field Goal for the Win activity that encourages students to model, explore and explain the dynamics of kicking a football through the uprights.

9. When you double a recipe, you’re going to need to understand ratios so your dinner guests don’t look like this.

10. before you take that family road trip , you’re going to want to calculate time and distance., 11. before you go candy shopping, you’re going to have to figure out x trick or treaters times x pieces of candy equals…, 12. if  you grow up to be an ice cream scientist, you’re going to have to understand the effect of temperature and pressure at the molecular level..

https://giphy.com/gifs/ice-lick-cream-3Z1kRYmLRQm5y

Explore states of matter and the processes that change cow milk into a cone of delicious decadence with this Ice Cream, Cool Science activity .

13. Once you have little ones, you’ll need to know how many diapers to buy for the month.

14. because what if it’s your turn to organize the annual ping pong tournament, and there are 7 players at a club with 4 tables, where each player plays against each other player, 15. when dressing for the day, you might want to consider the percent likelihood of rain., 16. if you go into medical research, you’re going to have to know how to solve equations..

Learn more about inspiring careers that improve lives with STEM Behind Health , a series of free activities from TI.

17. Understanding percentages will help you get the best deal at the mall. For example, how much will something cost with 40% off? What about once the 8% tax is added? What if it’s advertised as half-off?

https://giphy.com/gifs/blue-kawaii-pink-5aplc3D2G0IrC

18. Budgeting for vacation will require figuring out how many hours at your pay rate you’ll have to work to afford the trip you want.

19. when you volunteer to host the company holiday party, you’ll need to figure out how much food to get., 20. if you grow up to be a super villain, you’re going to need to use math to determine the most effective way to slow down the superhero and keep him from saving the day..

Put your students in the role of an arch-villain’s minions with Science Friction, a STEM Behind Hollywood activity .

21. You’ll definitely want to understand how to budget your money so you don’t look like this at the grocery checkout.

22. if you don’t work the numbers out in advance, you might at some point regret choosing that expensive out-of-state college., 23. before taking on a building project, remember the old saying—measure twice, cut once., 24. if have aspirations of being a fashion designer, you’ll have to understand geometry in order to make the perfect twirling skirt.

https://giphy.com/gifs/loop-bunny-ballet-yarFJggnH24da

Geometry and fashion design intersect in this STEM Behind Cool Careers activity .

25. Everyone loves a good bargain! Figuring out the best deal is not only fun, it’s smart!

26. if you can’t manage calculations, running the numbers at the car dealership might leave you feeling like this:, you might also like.

27 Essential Math Strategies for Teaching Students of All Ages

Even veteran teachers need to read these. Continue Reading

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5 Real Life Algebra Problems That You Solve Everyday

Algebra has a reputation for not being very useful in daily life. In fact, in my experience as a high school math teacher, the complaint that I get the most often is that we don’t spend enough time solving real life algebra problems.

You might be surprised to hear that I understand the frustration that my students experience. Unless we are solving real life algebra problems related to money in some way, algebra can feel very “artificial” or disconnected from real life.

My goal here is to walk you through 5 real life algebra problems that will give you a whole new appreciation for the application of algebra to the real-world. I am excited to help you see how many algebraic equations and algebraic concepts are applicable beyond just algebra word problems in your math class!

What is an Example of Algebra in Real Life?

While it is often seen as an abstract branch of mathematics, there are many real-life applications of algebra in everyday life. Now, it is unlikely that you will be solving quadratic equations while walking your dog, or solving real-world problems with linear equations while you play video games. But you can see examples of real life algebra problems all around you!

A simple example is when you want to quickly determine the total cost of a product including taxes, or the total cost after a discount from the original price. Knowing the total amount of money something will cost is a real-life scenario that everyone can relate to!

Depending on your chosen career path, you may see the use of algebra more often than others (I know I see it a lot in my daily life as a math teacher…!).

For example, if you are a business owner, you may use algebra to determine the number of labor hours to spread amongst your staff, or the lowest price you can sell your product for to break even.

For more uses of algebra, check out my list of 20  examples of algebra in real life !

What is an Example of an Algebra Problem in Real Life?

An algebra problem is a mathematical problem that requires the use of algebraic concepts and strategies to determine unknown values or unknown variables. Much like how the order of operations are required to evaluate numerical expressions, algebra problems require the problem solver to apply a set of rules in order to arrive at a solution.

Real world problems that require the use of algebra usually involve modelling real-life situations with  algebraic formulas . A formula is a specific equation that can be applied to solve a problem. Formulas make it possible to make predictions about a given real-life scenario.

For example, consider the following problem:

You are saving up for a new smartphone and currently have $200 in your savings account. Your plan is to save a certain amount of money each week from your allowance. If the smartphone costs $600, and you want to have enough money to buy it in 8 weeks, how much money should you save each week?

To solve this problem, we first need to use the information provided in the problem to create an equation that models the real-life scenario. Thinking about the problem in terms of variables, we can define T as the total of the savings, and variable x as the amount saved each week.

Since we know that we have a fixed value of 200, we can use the following equation to model this real world problem:

$$T=200+8x$$

This equation says “the total saved is equal to the original $200 plus whatever amount is saved per week, for 8 weeks”.

Substituting the total of the smartphone allows us to begin solving for the unknown variable x. Remember, when solving algebraic equations, you must apply the same operation to both sides of the equation.

$$ \begin{split} T&=200+8x  \\ \\ 600&=200+8x  \\ \\ 600-200 &= 8x \\ \\ 400 &= 8x \\ \\ \frac{400}{8} &= \frac{8x}{8} \\\\ 50 &= x \end{split} $$

Therefore, since x = 50, you should save $50 each week in order to save enough money for the smartphone. For more practice with the algebra used in this solution, check out this free collection of  solving two step equations worksheets !

5 Real Life Algebra Problems with Step-By-Step Solutions

There are so many real-life examples of algebra problems, but I want to focus on 5 here that I believe will convince you of just how applicable algebra is to the real-world! So let’s dig into these 5 real-world algebraic word problems!

Example #1: Comparing Cell Phone Plans

Link is considering two different cell phone plans. Plan A charges a monthly fee of $30 and an additional $0.10 per minute of talk time. Plan B charges a monthly fee of $45 regardless of how much time is used talking. How many minutes of total time talking will make the plans equal in cost?

The best way to start this problem is by writing two equations to represent each scenario. If C represents total cost, and x represents minutes of talk time used, the equations can be written as follows:

• Plan A: \(C=30+0.1x\)
• Plan B: \(C=45\)

Setting the first equation equal to the second equation will allow us to employ algebra to solve for the number of minutes that makes the two plans equal.

$$ \begin{split}  30+0.1x&=45 \\ \\ 30-30+0.1x&=45-30 \\ \\ 0.1x&=15 \\ \\ \frac{0.1x}{0.1}&=\frac{15}{0.1} \\ \\ x&=150 \end{split} $$

Therefore, the two cell phone plans are equal when 150 minutes of total time talking are used.

Example #2: Calculating Gallons of Gas

Zelda is driving from Hyrule to the Mushroom Kingdom, which are 180 miles apart. Her car can travel 30 miles per gallon of gas. Write an equation to represent the number of gallons of gas, G, that Zelda needs for the trip in terms of the distance, d, she needs to travel. Then calculate how many gallons of gas she needs for this trip.

The number of gallons of gas (G) Zelda needs for any trip can be represented by the equation \(G = \frac{d}{30}\). Since the distance between Hyrule and the Mushroom Kingdom is 180 miles, we can substitute 180 into the equation for  d  to determine the number of gallons of gas needed:

$$G=\frac{180}{30}=6$$

Therefore, Zelda needs 6 gallons of gas for her trip.

Example #3: Basketball Players in Action!

A basketball player shoots a basketball from a height of 6 feet above the ground. Unfortunately he completely misses the net and the ball bounces off court. A sports analyst models the path of the basketball using the equation \(h(t) = -16t^2 + 16t + 6\), where h(t) represents the height of the basketball above the ground in feet at time t seconds after the shot. Determine the time it takes for the basketball to hit the ground.

Since we are asked for when the ball hits the ground and  h(t)  is given as the height above the ground, we know that we are looking for the x-intercepts of this quadratic function. We therefore set the equation equal to zero and solve for x. 

Note that we cannot use  trinomial factoring  here since the quadratic is not factorable! Thankfully quadratic equations are solvable using the quadratic formula!

$$ \begin{split}  x&=\frac{-b \pm \sqrt{b^2-4ac}}{2a} \\\\ &=\frac{-16 \pm \sqrt{16^2-4(-16)(6)}}{2(-16)}\\\\ &=\frac{-16 \pm \sqrt{640}}{-32}\\\\ x&=-0.291 \\\\ x&=1.291 \end{split}$$

Therefore, the ball hits the ground after approximately 1.3 seconds. Remember that time cannot be negative, so the first answer is inadmissible and rejected!

Example #4: Saving for a Computer Game

You are saving to buy a new computer game that costs $90. You decide to save up for the computer game by depositing some money into a savings account that earns an annual interest rate of 5% (compounded monthly). You start with an initial deposit of $30 and plan to save for 22 months. Will you have enough to purchase the computer game?

This is an example of a math problem that connects to financial problems people encounter everyday! Since the account you chose earns  interest , we can apply a compound interest formula to help us out here:

$$A=P(1+i)^n$$

In this formula:

• A(t)  is the total amount of money.
• P  is the initial deposit (which is $30 in this case).
• i  is the monthly interest rate (5% annual interest, compounded monthly means that  i  is approximately 0.004167).
• n  is the time that has elapsed (since we are working with months, we multiply by 12)

We can set up our equation and see if our total amount of money is greater than $90:

$$\begin{split}  A(22)&=30(1.004167)^{22 \times 12} \\\\ &=$89.93 \end{split} $$

Remember to always include a dollar sign in your answer and to round to two decimal places when working with money!

Since our answer is approximately equal to $90, we can say that you will have enough money after 22 months! It’s time to get saving!

Example #5: How Many Tickets Did the Movie Theater Sell?

A movie theater charges $10 per ticket for adults and $6 per ticket for children. On a particular day, the theater sold a total of 150 tickets, and the total revenue for the day was $1350. Write a system of equations to represent this real-life scenario and then solve for the number of adult and child tickets sold.

Let’s assume that variable  x  represents the number of adult tickets sold and variable  y  represents the number of child tickets sold. We can set up two linear equations as follows:

• First Equation (the total number of tickets sold): \(x+y=150\) 
• Second Equation (the total revenue from ticket sales is 1350): \(10x+6y=1350\) 

We can use substitution to solve this linear system by rearranging the first equation and substituting it into the second equation. You can catch a quick overview of the substitution process by checking out  this substitution video  on my YouTube channel!

Rearranging the first equation into a different form to solve for  y  results in \(y=-x+150\). Substituting this expression for  y  into the second equation results in: 

$$ \begin{split}  10x+6(-x+150)&=1350 \\ \\ 10x-6x+900&=1350 \\ \\ 4x&=450\\ \\ x&=112.5\\ \\ \end{split}  $$

We then substitute this value for  x  into our expression for  y: 

$$ \begin{split}  y&=-x+150 \\ \\ &=-112.50+150 \\ \\ &=37.5\\ \\ \end{split}  $$

Since we can’t have fractional ticket sales, we can say that approximately 112 adult tickets were sold and 38 child tickets were sold.

Appreciating Real Life Algebra Problems

While algebra is often seen as an abstract topic, I am hopeful that I have shown you just how applicable it can be to real-life situations! Some of these examples you may have even encountered in your own life!

Even if you aren’t drawing up complex equations and solving them while you are playing basketball, combining basic math and problem solving is one of the most important skills people can have in both their work and their lives. 

I hope that I have helped you further your understanding of algebra, while growing an appreciation for the different ways it can be used in your own life!

Did you find this guide to real life algebra problems helpful? Share this post and subscribe to Math By The Pixel on YouTube for more helpful mathematics content!

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HYPOTHESIS AND THEORY article

Real world problem-solving.

• Human-Robot Interaction Laboratory, Department of Computer Science, Tufts University, Medford, MA, United States

Real world problem-solving (RWPS) is what we do every day. It requires flexibility, resilience, resourcefulness, and a certain degree of creativity. A crucial feature of RWPS is that it involves continuous interaction with the environment during the problem-solving process. In this process, the environment can be seen as not only a source of inspiration for new ideas but also as a tool to facilitate creative thinking. The cognitive neuroscience literature in creativity and problem-solving is extensive, but it has largely focused on neural networks that are active when subjects are not focused on the outside world, i.e., not using their environment. In this paper, I attempt to combine the relevant literature on creativity and problem-solving with the scattered and nascent work in perceptually-driven learning from the environment. I present my synthesis as a potential new theory for real world problem-solving and map out its hypothesized neural basis. I outline some testable predictions made by the model and provide some considerations and ideas for experimental paradigms that could be used to evaluate the model more thoroughly.

1. Introduction

In the Apollo 13 space mission, astronauts together with ground control had to overcome several challenges to bring the team safely back to Earth ( Lovell and Kluger, 2006 ). One of these challenges was controlling carbon dioxide levels onboard the space craft: “For 2 days straight [they] had worked on how to jury-rig the Odysseys canisters to the Aquarius's life support system. Now, using materials known to be available onboard the spacecraft—a sock, a plastic bag, the cover of a flight manual, lots of duct tape, and so on—the crew assembled a strange contraption and taped it into place. Carbon dioxide levels immediately began to fall into the safe range” ( Team, 1970 ; Cass, 2005 ).

The success of Apollo 13's recovery from failure is often cited as a glowing example of human resourcefulness and inventiveness alongside more well-known inventions and innovations over the course of human history. However, this sort of inventive capability is not restricted to a few creative geniuses, but an ability present in all of us, and exemplified in the following mundane example. Consider a situation when your only suit is covered in lint and you do not own a lint remover. You see a roll of duct tape, and being resourceful you reason that it might be a good substitute. You then solve the problem of lint removal by peeling a full turn's worth of tape and re-attaching it backwards onto the roll to expose the sticky side all around the roll. By rolling it over your suit, you can now pick up all the lint.

In both these examples (historic as well as everyday), we see evidence for our innate ability to problem-solve in the real world. Solving real world problems in real time given constraints posed by one's environment are crucial for survival. At the core of this skill is our mental capability to get out of “sticky situations” or impasses, i.e., difficulties that appear unexpectedly as impassable roadblocks to solving the problem at hand. But, what are the cognitive processes that enable a problem solver to overcome such impasses and arrive at a solution, or at least a set of promising next steps?

A central aspect of this type of real world problem solving, is the role played by the solver's surrounding environment during the problem-solving process. Is it possible that interaction with one's environment can facilitate creative thinking? The answer to this question seems somewhat obvious when one considers the most famous anecdotal account of creative problem solving, namely that of Archimedes of Syracuse. During a bath, he found a novel way to check if the King's crown contained non-gold impurities. The story has traditionally been associated with the so-called “Eureka moment,” the sudden affective experience when a solution to a particularly thorny problem emerges. In this paper, I want to temporarily turn our attention away from the specific “aha!” experience itself and take particular note that Archimedes made this discovery, not with his eyes closed at a desk, but in a real-world context of a bath 1 . The bath was not only a passive, relaxing environment for Archimedes, but also a specific source of inspiration. Indeed it was his noticing the displacement of water that gave him a specific methodology for measuring the purity of the crown; by comparing how much water a solid gold bar of the same weight would displace as compared with the crown. This sort of continuous environmental interaction was present when the Apollo 13 engineers discovered their life-saving solution, and when you solved the suit-lint-removal problem with duct tape.

The neural mechanisms underlying problem-solving have been extensively studied in the literature, and there is general agreement about the key functional networks and nodes involved in various stages of problem-solving. In addition, there has been a great deal of work in studying the neural basis for creativity and insight problem solving, which is associated with the sudden emergence of solutions. However, in the context of problem-solving, creativity, and insight have been researched as largely an internal process without much interaction with and influence from the external environment ( Wegbreit et al., 2012 ; Abraham, 2013 ; Kounios and Beeman, 2014 ) 2 . Thus, there are open questions of what role the environment plays during real world problem-solving (RWPS) and how the brain enables the assimilation of novel items during these external interactions.

In this paper, I synthesize the literature on problem-solving, creativity and insight, and particularly focus on how the environment can inform RWPS. I explore three environmentally-informed mechanisms that could play a critical role: (1) partial-cue driven context-shifting, (2) heuristic prototyping and learning novel associations, and (3) learning novel physical inferences. I begin first with some intuitions about real world problem solving, that might help ground this discussion and providing some key distinctions from more traditional problem solving research. Then, I turn to a review of the relevant literature on problem-solving, creativity, and insight first, before discussing the three above-mentioned environmentally-driven mechanisms. I conclude with a potential new model and map out its hypothesized neural basis.

2. Problem Solving, Creativity, and Insight

2.1. what is real world problem-solving.

Archimedes was embodied in the real world when he found his solution. In fact, the real world helped him solve the problem. Whether or not these sorts of historic accounts of creative inspiration are accurate 3 , they do correlate with some of our own key intuitions about how problem solving occurs “in the wild.” Real world problem solving (RWPS) is different from those that occur in a classroom or in a laboratory during an experiment. They are often dynamic and discontinuous, accompanied by many starts and stops. Solvers are never working on just one problem. Instead, they are simultaneously juggling several problems of varying difficulties and alternating their attention between them. Real world problems are typically ill-defined, and even when they are well-defined, often have open-ended solutions. Coupled with that is the added aspect of uncertainty associated with the solver's problem solving strategies. As introduced earlier, an important dimension of RWPS is the continuous interaction between the solver and their environment. During these interactions, the solver might be inspired or arrive at an “aha!” moment. However, more often than not, the solver experiences dozens of minor discovery events— “hmmm, interesting…” or “wait, what?…” moments. Like discovery events, there's typically never one singular impasse or distraction event. The solver must iterate through the problem solving process experiencing and managing these sorts of intervening events (including impasses and discoveries). In summary, RWPS is quite messy and involves a tight interplay between problem solving, creativity, and insight. Next, I explore each of these processes in more detail and explicate a possible role of memory, attention, conflict management and perception.

2.2. Analytical Problem-Solving

In psychology and neuroscience, problem-solving broadly refers to the inferential steps taken by an agent 4 that leads from a given state of affairs to a desired goal state ( Barbey and Barsalou, 2009 ). The agent does not immediately know how this goal can be reached and must perform some mental operations (i.e., thinking) to determine a solution ( Duncker, 1945 ).

The problem solving literature divides problems based on clarity (well-defined vs. ill-defined) or on the underlying cognitive processes (analytical, memory retrieval, and insight) ( Sprugnoli et al., 2017 ). While memory retrieval is an important process, I consider it as a sub-process to problem solving more generally. I first focus on analytical problem-solving process, which typically involves problem-representation and encoding, and the process of forming and executing a solution plan ( Robertson, 2016 ).

2.2.1. Problem Definition and Representation

An important initial phase of problem-solving involves defining the problem and forming a representation in the working memory. During this phase, components of the prefrontal cortex (PFC), default mode network (DMN), and the dorsal anterior cingulate cortex (dACC) have been found to be activated. If the problem is familiar and well-structured, top-down executive control mechanisms are engaged and the left prefrontal cortex including the frontopolar, dorso-lateral (dlPFC), and ventro-lateral (vlPFC) are activated ( Barbey and Barsalou, 2009 ). The DMN along with the various structures in the medial temporal lobe (MTL) including the hippocampus (HF), parahippocampal cortex, perirhinal and entorhinal cortices are also believed to have limited involvement, especially in episodic memory retrieval activities during this phase ( Beaty et al., 2016 ). The problem representation requires encoding problem information for which certain visual and parietal areas are also involved, although the extent of their involvement is less clear ( Anderson and Fincham, 2014 ; Anderson et al., 2014 ).

2.2.1.1. Working memory

An important aspect of problem representation is the engagement and use of working memory (WM). The WM allows for the maintenance of relevant problem information and description in the mind ( Gazzaley and Nobre, 2012 ). Research has shown that WM tasks consistently recruit the dlPFC and left inferior frontal cortex (IC) for encoding an manipulating information; dACC for error detection and performance adjustment; and vlPFC and the anterior insula (AI) for retrieving, selecting information and inhibitory control ( Chung and Weyandt, 2014 ; Fang et al., 2016 ).

2.2.1.2. Representation

While we generally have a sense for the brain regions that are functionally influential in problem definition, less is known about how exactly events are represented within these regions. One theory for how events are represented in the PFC is the structured event complex theory (SEC), in which components of the event knowledge are represented by increasingly higher-order convergence zones localized within the PFC, akin to the convergence zones (from posterior to anterior) that integrate sensory information in the brain ( Barbey et al., 2009 ). Under this theory, different zones in the PFC (left vs. right, anterior vs. posterior, lateral vs. medial, and dorsal vs. ventral) represent different aspects of the information contained in the events (e.g., number of events to be integrated together, the complexity of the event, whether planning, and action is needed). Other studies have also suggested the CEN's role in tasks requiring cognitive flexibility, and functions to switch thinking modes, levels of abstraction of thought and consider multiple concepts simultaneously ( Miyake et al., 2000 ).

Thus, when the problem is well-structured, problem representation is largely an executive control activity coordinated by the PFC in which problem information from memory populates WM in a potentially structured representation. Once the problem is defined and encoded, planning and execution of a solution can begin.

2.2.2. Planning

The central executive network (CEN), particularly the PFC, is largely involved in plan formation and in plan execution. Planning is the process of generating a strategy to advance from the current state to a goal state. This in turn involves retrieving a suitable solution strategy from memory and then coordinating its execution.

2.2.2.1. Plan formation

The dlPFC supports sequential planning and plan formation, which includes the generation of hypothesis and construction of plan steps ( Barbey and Barsalou, 2009 ). Interestingly, the vlPFC and the angular gyrus (AG), implicated in a variety of functions including memory retrieval, are also involved in plan formation ( Anderson et al., 2014 ). Indeed, the AG together with the regions in the MTL (including the HF) and several other regions form a what is known as the “core” network. The core network is believed to be activated when recalling past experiences, imagining fictitious, and future events and navigating large-scale spaces ( Summerfield et al., 2010 ), all key functions for generating plan hypotheses. A recent study suggests that the AG is critical to both episodic simulation, representation, and episodic memory ( Thakral et al., 2017 ). One possibility for how plans are formulated could involve a dynamic process of retrieving an optimal strategy from memory. Research has shown significant interaction between striatal and frontal regions ( Scimeca and Badre, 2012 ; Horner et al., 2015 ). The striatum is believed to play a key role in declarative memory retrieval, and specifically helping retrieve optimal (or previously rewarded) memories ( Scimeca and Badre, 2012 ). Relevant to planning and plan formation, Scimeca & Badre have suggested that the striatum plays two important roles: (1) in mapping acquired value/utility to action selection, and thereby helping plan formation, and (2) modulation and re-encoding of actions and other plan parameters. Different types of problems require different sets of specialized knowledge. For example, the knowledge needed to solve mathematical problems might be quite different (albeit overlapping) from the knowledge needed to select appropriate tools in the environment.

Thus far, I have discussed planning and problem representation as being domain-independent, which has allowed me to outline key areas of the PFC, MTL, and other regions relevant to all problem-solving. However, some types of problems require domain-specific knowledge for which other regions might need to be recruited. For example, when planning for tool-use, the superior parietal lobe (SPL), supramarginal gyrus (SMG), anterior inferior parietal lobe (AIPL), and certain portions of the temporal and occipital lobe involved in visual and spatial integration have been found to be recruited ( Brandi et al., 2014 ). It is believed that domain-specific information stored in these regions is recovered and used for planning.

2.2.2.2. Plan execution

Once a solution plan has been recruited from memory and suitably tuned for the problem on hand, the left-rostral PFC, caudate nucleus (CN), and bilateral posterior parietal cortices (PPC) are responsible for translating the plan into executable form ( Stocco et al., 2012 ). The PPC stores and maintains “mental template” of the executable form. Hemispherical division of labor is particularly relevant in planning where it was shown that when planning to solve a Tower of Hanoi (block moving) problem, the right PFC is involved in plan construction whereas the left PFC is involved in controlling processes necessary to supervise the execution of the plan ( Newman and Green, 2015 ). On a separate note and not the focus of this paper, plan execution and problem-solving can require the recruitment of affective and motivational processing in order to supply the agent with the resolve to solve problems, and the vmPFC has been found to be involved in coordinating this process ( Barbey and Barsalou, 2009 ).

2.3. Creativity

During the gestalt movement in the 1930s, Maier noted that “most instances of “real” problem solving involves creative thinking” ( Maier, 1930 ). Maier performed several experiments to study mental fixation and insight problem solving. This close tie between insight and creativity continues to be a recurring theme, one that will be central to the current discussion. If creativity and insight are linked to RWPS as noted by Maier, then it is reasonable to turn to the creativity and insight literature for understanding the role played by the environment. A large portion of the creativity literature has focused on viewing creativity as an internal process, one in which the solvers attention is directed inwards, and toward internal stimuli, to facilitate the generation of novel ideas and associations in memory ( Beaty et al., 2016 ). Focusing on imagination, a number of researchers have looked at blinking, eye fixation, closing eyes, and looking nowhere behavior and suggested that there is a shift of attention from external to internal stimuli during creative problem solving ( Salvi and Bowden, 2016 ). The idea is that shutting down external stimuli reduces cognitive load and focuses attention internally. Other experiments studying sleep behavior have also noted the beneficial role of internal stimuli in problem solving. The notion of ideas popping into ones consciousness, suddenly, during a shower is highly intuitive for many and researchers have attempted to study this phenomena through the lens of incubation, and unconscious thought that is internally-driven. There have been several theories and counter-theories proposed to account specifically for the cognitive processes underlying incubation ( Ritter and Dijksterhuis, 2014 ; Gilhooly, 2016 ), but none of these theories specifically address the role of the external environment.

The neuroscience of creativity has also been extensively studied and I do not focus on an exhaustive literature review in this paper (a nice review can be found in Sawyer, 2011 ). From a problem-solving perspective, it has been found that unlike well-structured problems, ill-structured problems activate the right dlPFC. Most of the past work on creativity and creative problem-solving has focused on exploring memory structures and performing internally-directed searches. Creative idea generation has primarily been viewed as internally directed attention ( Jauk et al., 2012 ; Benedek et al., 2016 ) and a primary mechanism involved is divergent thinking , which is the ability to produce a variety of responses in a given situation ( Guilford, 1962 ). Divergent thinking is generally thought to involve interactions between the DMN, CEN, and the salience network ( Yoruk and Runco, 2014 ; Heinonen et al., 2016 ). One psychological model of creative cognition is the Geneplore model that considers two major phases of generation (memory retrieval and mental synthesis) and exploration (conceptual interpretation and functional inference) ( Finke et al., 1992 ; Boccia et al., 2015 ). It has been suggested that the associative mode of processing to generate new creative association is supported by the DMN, which includes the medial PFC, posterior cingulate cortex (PCC), tempororparietal juntion (TPJ), MTL, and IPC ( Beaty et al., 2014 , 2016 ).

That said, the creativity literature is not completely devoid of acknowledging the role of the environment. In fact, it is quite the opposite. Researchers have looked closely at the role played by externally provided hints from the time of the early gestalt psychologists and through to present day studies ( Öllinger et al., 2017 ). In addition to studying how hints can help problem solving, researchers have also looked at how directed action can influence subsequent problem solving—e.g., swinging arms prior to solving the two-string puzzle, which requires swinging the string ( Thomas and Lleras, 2009 ). There have also been numerous studies looking at how certain external perceptual cues are correlated with creativity measures. Vohs et al. suggested that untidiness in the environment and the increased number of potential distractions helps with creativity ( Vohs et al., 2013 ). Certain colors such as blue have been shown to help with creativity and attention to detail ( Mehta and Zhu, 2009 ). Even environmental illumination, or lack thereof, have been shown to promote creativity ( Steidle and Werth, 2013 ). However, it is important to note that while these and the substantial body of similar literature show the relationship of the environment to creative problem solving, they do not specifically account for the cognitive processes underlying the RWPS when external stimuli are received.

2.4. Insight Problem Solving

Analytical problem solving is believed to involve deliberate and conscious processing that advances step by step, allowing solvers to be able to explain exactly how they solved it. Inability to solve these problems is often associated with lack of required prior knowledge, which if provided, immediately makes the solution tractable. Insight, on the other hand, is believed to involve a sudden and unexpected emergence of an obvious solution or strategy sometimes accompanied by an affective aha! experience. Solvers find it difficult to consciously explain how they generated a solution in a sequential manner. That said, research has shown that having an aha! moment is neither necessary nor sufficient to insight and vice versa ( Danek et al., 2016 ). Generally, it is believed that insight solvers acquire a full and deep understanding of the problem when they have solved it ( Chu and Macgregor, 2011 ). There has been an active debate in the problem solving community about whether insight is something special. Some have argued that it is not, and that there are no special or spontaneous processes, but simply a good old-fashioned search of a large problem space ( Kaplan and Simon, 1990 ; MacGregor et al., 2001 ; Ash and Wiley, 2006 ; Fleck, 2008 ). Others have argued that insight is special and suggested that it is likely a different process ( Duncker, 1945 ; Metcalfe, 1986 ; Kounios and Beeman, 2014 ). This debate lead to two theories for insight problem solving. MacGregor et al. proposed the Criterion for Satisfactory Progress Theory (CSPT), which is based on Newell and Simons original notion of problem solving as being a heuristic search through the problem space ( MacGregor et al., 2001 ). The key aspect of CSPT is that the solver is continually monitoring their progress with some set of criteria. Impasses arise when there is a criterion failure, at which point the solver tries non-maximal but promising states. The representational change theory (RCT) proposed by Ohlsson et al., on the other hand, suggests that impasses occur when the goal state is not reachable from an initial problem representation (which may have been generated through unconscious spreading activation) ( Ohlsson, 1992 ). In order to overcome an impasse, the solver needs to restructure the problem representation, which they can do by (1) elaboration (noticing new features of a problem), (2) re-encoding fixing mistaken or incomplete representations of the problem, and by (3) changing constraints. Changing constraints is believed to involve two sub-processes of constraint relaxation and chunk-decomposition.

The current position is that these two theories do not compete with each other, but instead complement each other by addressing different stages of problem solving: pre- and post-impasse. Along these lines, Ollinger et al. proposed an extended RCT (eRCT) in which revising the search space and using heuristics was suggested as being a dynamic and iterative and recursive process that involves repeated instances of search, impasse and representational change ( Öllinger et al., 2014 , 2017 ). Under this theory, a solver first forms a problem representation and begins searching for solutions, presumably using analytical problem solving processes as described earlier. When a solution cannot be found, the solver encounters an impasse, at which point the solver must restructure or change the problem representation and once again search for a solution. The model combines both analytical problem solving (through heuristic searches, hill climbing and progress monitoring), and creative mechanisms of constraint relaxation and chunk decomposition to enable restructuring.

Ollingers model appears to comprehensively account for both analytical and insight problem solving and, therefore, could be a strong candidate to model RWPS. However, while compelling, it is nevertheless an insufficient model of RWPS for many reasons, of which two are particularly significant for the current paper. First, the model does explicitly address mechanisms by which external stimuli might be assimilated. Second, the model is not sufficiently flexible to account for other events (beyond impasse) occurring during problem solving, such as distraction, mind-wandering and the like.

So, where does this leave us? I have shown the interplay between problem solving, creativity and insight. In particular, using Ollinger's proposal, I have suggested (maybe not quite explicitly up until now) that RWPS involves some degree of analytical problem solving as well as the post-impasse more creative modes of problem restructuring. I have also suggested that this model might need to be extended for RWPS along two dimensions. First, events such as impasses might just be an instance of a larger class of events that intervene during problem solving. Thus, there needs to be an accounting of the cognitive mechanisms that are potentially influenced by impasses and these other intervening events. It is possible that these sorts of events are crucial and trigger a switch in attentional focus, which in turn facilitates switching between different problem solving modes. Second, we need to consider when and how externally-triggered stimuli from the solver's environment can influence the problem solving process. I detail three different mechanisms by which external knowledge might influence problem solving. I address each of these ideas in more detail in the next two sections.

3. Event-Triggered Mode Switching During Problem-Solving

3.1. impasse.

When solving certain types of problems, the agent might encounter an impasse, i.e., some block in its ability to solve the problem ( Sprugnoli et al., 2017 ). The impasse may arise because the problem may have been ill-defined to begin with causing incomplete and unduly constrained representations to have been formed. Alternatively, impasses can occur when suitable solution strategies cannot be retrieved from memory or fail on execution. In certain instances, the solution strategies may not exist and may need to be generated from scratch. Regardless of the reason, an impasse is an interruption in the problem solving process; one that was running conflict-free up until the point when a seemingly unresolvable issue or an error in the predicted solution path was encountered. Seen as a conflict encountered in the problem-solving process it activates the anterior cingulate cortex (ACC). It is believed that the ACC not only helps detect the conflict, but also switch modes from one of “exploitation” (planning) to “exploration” (search) ( Quilodran et al., 2008 ; Tang et al., 2012 ), and monitors progress during resolution ( Chu and Macgregor, 2011 ). Some mode switching duties are also found to be shared with the AI (the ACC's partner in the salience network), however, it is unclear exactly the extent of this function-sharing.

Even though it is debatable if impasses are a necessary component of insight, they are still important as they provide a starting point for the creativity ( Sprugnoli et al., 2017 ). Indeed, it is possible that around the moment of impasse, the AI and ACC together, as part of the salience network play a crucial role in switching thought modes from analytical planning mode to creative search and discovery mode. In the latter mode, various creative mechanisms might be activated allowing for a solution plan to emerge. Sowden et al. and many others have suggested that the salience network is potentially a candidate neurobiological mechanism for shifting between thinking processes, more generally ( Sowden et al., 2015 ). When discussing various dual-process models as they relate to creative cognition, Sowden et al. have even noted that the ACC activation could be useful marker to identify shifting as participants work creative problems.

3.2. Defocused Attention

As noted earlier, in the presence of an impasse there is a shift from an exploitative (analytical) thinking mode to an exploratory (creative) thinking mode. This shift impacts several networks including, for example, the attention network. It is believed attention can switch between a focused mode and a defocused mode. Focused attention facilitates analytic thought by constraining activation such that items are considered in a compact form that is amenable to complex mental operations. In the defocused mode, agents expand their attention allowing new associations to be considered. Sowden et al. (2015) note that the mechanism responsible for adjustments in cognitive control may be linked to the mechanisms responsible for attentional focus. The generally agreed position is that during generative thinking, unconscious cognitive processes activated through defocused attention are more prevalent, whereas during exploratory thinking, controlled cognition activated by focused attention becomes more prevalent ( Kaufman, 2011 ; Sowden et al., 2015 ).

Defocused attention allows agents to not only process different aspects of a situation, but to also activate additional neural structures in long term memory and find new associations ( Mendelsohn, 1976 ; Yoruk and Runco, 2014 ). It is believed that cognitive material attended to and cued by positive affective state results in defocused attention, allowing for more complex cognitive contexts and therefore a greater range of interpretation and integration of information ( Isen et al., 1987 ). High attentional levels are commonly considered a typical feature of highly creative subjects ( Sprugnoli et al., 2017 ).

4. Role of the Environment

In much of the past work the focus has been on treating creativity as largely an internal process engaging the DMN to assist in making novel connections in memory. The suggestion has been that “individual needs to suppress external stimuli and concentrate on the inner creative process during idea generation” ( Heinonen et al., 2016 ). These ideas can then function as seeds for testing and problem-solving. While true of many creative acts, this characterization does not capture how creative ideas arise in many real-world creative problems. In these types of problems, the agent is functioning and interacting with its environment before, during and after problem-solving. It is natural then to expect that stimuli from the environment might play a role in problem-solving. More specifically, it can be expected that through passive and active involvement with the environment, the agent is (1) able to trigger an unrelated, but potentially useful memory relevant for problem-solving, (2) make novel connections between two events in memory with the environmental cue serving as the missing link, and (3) incorporate a completely novel information from events occuring in the environment directly into the problem-solving process. I explore potential neural mechanisms for these three types of environmentally informed creative cognition, which I hypothesize are enabled by defocused attention.

4.1. Partial Cues Trigger Relevant Memories Through Context-Shifting

I have previously discussed the interaction between the MTL and PFC in helping select task-relevant and critical memories for problem-solving. It is well-known that pattern completion is an important function of the MTL and one that enables memory retrieval. Complementary Learning Theory (CLS) and its recently updated version suggest that the MTL and related structures support initial storage as well as retrieval of item and context-specific information ( Kumaran et al., 2016 ). According to CLS theory, the dentate gyrus (DG) and the CA3 regions of the HF are critical to selecting neural activity patterns that correspond to particular experiences ( Kumaran et al., 2016 ). These patterns might be distinct even if experiences are similar and are stabilized through increases in connection strengths between the DG and CA3. Crucially, because of the connection strengths, reactivation of part of the pattern can activate the rest of it (i.e., pattern completion). Kumaran et al. have further noted that if consistent with existing knowledge, these new experiences can be quickly replayed and interleaved into structured representations that form part of the semantic memory.

Cues in the environment provided by these experiences hold partial information about past stimuli or events and this partial information converges in the MTL. CLS accounts for how these cues might serve to reactivate partial patterns, thereby triggering pattern completion. When attention is defocused I hypothesize that (1) previously unnoticed partial cues are considered, and (2) previously noticed partial cues are decomposed to produce previously unnoticed sub-cues, which in turn are considered. Zabelina et al. (2016) have shown that real-world creativity and creative achievement is associated with “leaky attention,” i.e., attention that allows for irrelevant information to be noticed. In two experiments they systematically explored the relationship between two notions of creativity— divergent thinking and real-world creative achievement—and the use of attention. They found that attentional use is associated in different ways for each of the two notions of creativity. While divergent thinking was associated with flexible attention, it does not appear to be leaky. Instead, selective focus and inhibition components of attention were likely facilitating successful performance on divergent thinking tasks. On the other hand, real-world creative achievement was linked to leaky attention. RWPS involves elements of both divergent thinking and of real-world creative achievement, thus I would expect some amount of attentional leaks to be part of the problem solving process.

Thus, it might be the case that a new set of cues or sub-cues “leak” in and activate memories that may not have been previously considered. These cues serve to reactivate a diverse set of patterns that then enable accessing a wide range of memories. Some of these memories are extra-contextual, in that they consider the newly noticed cues in several contexts. For example, when unable to find a screwdriver, we might consider using a coin. It is possible that defocused attention allows us to consider the coin's edge as being a potentially relevant cue that triggers uses for the thin edge outside of its current context in a coin. The new cues (or contexts) may allow new associations to emerge with cues stored in memory, which can occur during incubation. Objects and contexts are integrated into memory automatically into a blended representation and changing contexts disrupts this recognition ( Hayes et al., 2007 ; Gabora, 2016 ). Cue-triggered context shifting allows an agent to break-apart a memory representation, which can then facilitate problem-solving in new ways.

4.2. Heuristic Prototyping Facilitates Novel Associations

It has long been the case that many scientific innovations have been inspired by events in nature and the surrounding environment. As noted earlier, Archimedes realized the relationship between the volume of an irregularly shaped object and the volume of water it displaced. This is an example of heuristic prototyping where the problem-solver notices an event in the environment, which then triggers the automatic activation of a heuristic prototype and the formation of novel associations (between the function of the prototype and the problem) which they can then use to solve the problem ( Luo et al., 2013 ). Although still in its relative infancy, there has been some recent research into the neural basis for heuristic prototyping. Heuristic prototype has generally been defined as an enlightening prototype event with a similar element to the current problem and is often composed of a feature and a function ( Hao et al., 2013 ). For example, in designing a faster and more efficient submarine hull, a heuristic prototype might be a shark's skin, while an unrelated prototype might be a fisheye camera ( Dandan et al., 2013 ).

Research has shown that activating the feature function of the right heuristic prototype and linking it by way of semantic similarity to the required function of the problem was the key mechanism people used to solve several scienitific insight problems ( Yang et al., 2016 ). A key region activated during heuristic prototyping is the dlPFC and it is believed to be generally responsible for encoding the events into memory and may play an important role in selecting and retrieving the matched unsolved technical problem from memory ( Dandan et al., 2013 ). It is also believed that the precuneus plays a role in automatic retrieval of heuristic information allowing the heuristic prototype and the problem to combine ( Luo et al., 2013 ). In addition to semantic processing, certain aspects of visual imagery have also been implicated in heuristic prototyping leading to the suggestion of the involvement of Broadman's area BA 19 in the occipital cortex.

There is some degree of overlap between the notions of heuristic prototyping and analogical transfer (the mapping of relations from one domain to another). Analogical transfer is believed to activate regions in the left medial fronto-parietal system (dlPFC and the PPC) ( Barbey and Barsalou, 2009 ). I suggest here that analogical reasoning is largely an internally-guided process that is aided by heuristic prototyping which is an externally-guided process. One possible way this could work is if heuristic prototyping mechanisms help locate the relevant memory with which to then subsequently analogize.

4.3. Making Physical Inferences to Acquire Novel Information

The agent might also be able to learn novel facts about their environment through passive observation as well as active experimentation. There has been some research into the neural basis for causal reasoning ( Barbey and Barsalou, 2009 ; Operskalski and Barbey, 2016 ), but beyond its generally distributed nature, we do not know too much more. Beyond abstract causal reasoning, some studies looked into the cortical regions that are activated when people watch and predict physical events unfolding in real-time and in the real-world ( Fischer et al., 2016 ). It was found that certain regions were associated with representing types of physical concepts, with the left intraparietal sulcus (IPS) and left middle frontal gyrus (MFG) shown to play a role in attributing causality when viewing colliding objects ( Mason and Just, 2013 ). The parahippocampus (PHC) was associated with linking causal theory to observed data and the TPJ was involved in visualizing movement of objects and actions in space ( Mason and Just, 2013 ).

5. Proposed Theory

I noted earlier that Ollinger's model for insight problem solving, while serving as a good candidate for RWPS, requires extension. In this section, I propose a candidate model that includes some necessary extensions to Ollinger's framework. I begin by laying out some preliminary notions that underlie the proposed model.

5.1. Dual Attentional Modes

I propose that the attention-switching mechanism described earlier is at the heart of RWPS and enables two modes of operation: focused and defocused mode. In the focused mode, the problem representation is more or less fixed, and problem solving proceeds in a focused and goal directed manner through search, planning, and execution mechanisms. In the defocused mode, problem solving is not necessarily goal directed, but attempts to generate ideas, driven by both internal and external items.

At first glance, these modes might seem similar to convergent and divergent thinking modes postulated by numerous others to account for creative problem solving. Divergent thinking allows for the generation of new ideas and convergent thinking allows for verification and selection of generated ideas. So, it might seem that focused mode and convergent thinking are similar and likewise divergent and defocused mode. They are, however, quite different. The modes relate less to idea generation and verification, and more to the specific mechanisms that are operating with regard to a particular problem at a particular moment in time. Convergent and divergent processes may be occurring during both defocused and focused modes. Some degree of divergent processes may be used to search and identify specific solution strategies in focused mode. Also, there might be some degree of convergent idea verification occuring in defocused mode as candidate items are evaluated for their fit with the problem and goal. Thus, convergent and divergent thinking are one amongst many mechanisms that are utilized in focused and defocused mode. Each of these two modes has to do with degree of attention placed on a particular problem.

There have been numerous dual-process and dual-systems models of cognition proposed over the years. To address criticisms raised against these models and to unify some of the terminology, Evans & Stanovich proposed a dual-process model comprising Type 1 and Type 2 thought ( Evans and Stanovich, 2013 ; Sowden et al., 2015 ). Type 1 processes are those that are believed to be autonomous and do not require working memory. Type 2 processes, on the other hand, are believed to require working memory and are cognitively decoupled to prevent real-world representations from becoming confused with mental simulations ( Sowden et al., 2015 ). While acknowledging various other attributes that are often used to describe dual process models (e.g., fast/slow, associative/rule-based, automatic/controlled), Evans & Stanovich note that these attributes are merely frequent correlates and not defining characteristics of Type 1 or Type 2 processes. The proposed dual attentional modes share some similarities with the Evans & Stanovich Type 1 and 2 models. Specifically, Type 2 processes might occur in focused attentional mode in the proposed model as they typically involve the working memory and certain amount of analytical thought and planning. Similarly, Type 1 processes are likely engaged in defocused attentional mode as there are notions of associative and generative thinking that might be facilitated when attention has been defocused. The crucial difference between the proposed model and other dual-process models is that the dividing line between focused and defocused attentional modes is the degree of openness to internal and external stimuli (by various networks and functional units in the brain) when problem solving. Many dual process models were designed to classify the “type” of thinking process or a form of cognitive processing. In some sense, the “processes” in dual process theories are characterized by the type of mechanism of operation or the type of output they produced. Here, I instead characterize and differentiate the modes of thinking by the receptivity of different functional units in the brain to input during problem solving.

This, however, raises a different question of the relationship between these attentional modes and conscious vs. unconscious thinking. It is clear that both the conscious and unconscious are involved in problem solving, as well as in RWPS. Here, I claim that a problem being handled is, at any given point in time, in either a focused mode or in a defocused mode. When in the focused mode, problem solving primarily proceeds in a manner that is available for conscious deliberation. More specifically, problem space elements and representations are tightly managed and plans and strategies are available in the working memory and consciously accessible. There are, however, secondary unconscious operations in the focused modes that includes targeted memory retrieval and heuristic-based searches. In the defocused mode, the problem is primarily managed in an unconscious way. The problem space elements are broken apart and loosely managed by various mechanisms that do not allow for conscious deliberation. That said, it is possible that some problem parameters remain accessible. For example, it is possible that certain goal information is still maintained consciously. It is also possible that indexes to all the problems being considered by the solver are maintained and available to conscious awareness.

5.2. RWPS Model

Returning to Ollinger's model for insight problem solving, it now becomes readily apparent how this model can be modified to incorporate environmental effects as well as generalizing the notion of intervening events beyond that of impasses. I propose a theory for RWPS that begins with standard analytical problem-solving process (See Figures 1 , 2 ).

Figure 1 . Summary of neural activations during focused problem-solving (Left) and defocused problem-solving (Right) . During defocused problem-solving, the salience network (insula and ACC) coordinates the switching of several networks into a defocused attention mode that permits the reception of a more varied set of stimuli and interpretations via both the internally-guided networks (default mode network DMN) and externally guided networks (Attention). PFC, prefrontal cortex; ACC, anterior cingulate cortex; PCC, posterior cingulate cortex; IPC, inferior parietal cortex; PPC, posterior parietal cortex; IPS, intra-parietal sulcus; TPJ, temporoparietal junction; MTL, medial temporal lobe; FEF, frontal eye field.

Figure 2 . Proposed Model for Real World Problem Solving (RWPS). The corresponding neural correlates are shown in italics. During problem-solving, an initial problem representation is formed based on prior knowledge and available perceptual information. The problem-solving then proceeds in a focused, goal-directed mode until the goal is achieved or a defocusing event (e.g., impasse or distraction) occurs. During focused mode operation, the solver interacts with the environment in directed manner, executing focused plans, and allowing for predicted items to be activated by the environment. When a defocusing event occurs, the problem-solving then switches into a defocused mode until a focusing event (e.g., discovery) occurs. In defocused mode, the solver performs actions unrelated to the problem (or is inactive) and is receptive to a set of environmental triggers that activate novel aspects using the three mechanisms discussed in this paper. When a focusing event occurs, the diffused problem elements cohere into a restructured representation and problem-solving returns into a focused mode.

5.2.1. Focused Problem Solving Mode

Initially, both prior knowledge and perceptual entities help guide the creation of problem representations in working memory. Prior optimal or rewarding solution strategies are obtained from LTM and encoded in the working memory as well. This process is largely analytical and the solver interacts with their environment through focused plan or idea execution, targeted observation of prescribed entities, and estimating prediction error of these known entities. More specifically, when a problem is presented, the problem representations are activated and populated into working memory in the PFC, possibly in structured representations along convergence zones. The PFC along with the Striatum and the MTL together attempt at retrieving an optimal or previously rewarded solution strategy from long term memory. If successfully retrieved, the solution strategy is encoded into the PPC as a mental template, which then guides relevant motor control regions to execute the plan.

5.2.2. Defocusing Event-Triggered Mode Switching

The search and solve strategy then proceeds analytically until a “defocusing event” is encountered. The salience network (AI and ACC) monitor for conflicts and attempt to detect any such events in the problem-solving process. As long as no conflicts are detected, the salience network focuses on recruiting networks to achieve goals and suppresses the DMN ( Beaty et al., 2016 ). If the plan execution or retrieval of the solution strategy fails, then a defocusing event is detected and the salience network performs mode switching. The salience network dynamically switches from the focused problem-solving mode to a defocused problem-solving mode ( Menon, 2015 ). Ollinger's current model does not account for other defocusing events beyond an impasse, but it is not inconceivable that there could be other such events triggered by external stimuli (e.g., distraction or an affective event) or by internal stimuli (e.g., mind wandering).

5.2.3. Defocused Problem Solving Mode

In defocused mode, the problem is operated on by mechanisms that allow for the generation and testing of novel ideas. Several large-scale brain networks are recruited to explore and generate new ideas. The search for novel ideas is facilitated by generally defocused attention, which in turn allows for creative idea generation from both internal as well as external sources. The salience network switches operations from defocused event detection to focused event or discovery detection, whereby for example, environmental events or ideas that are deemed interesting can be detected. During this idea exploration phase, internally, the DMN is no longer suppressed and attempts to generate new ideas for problem-solving. It is known that the IPC is involved in the generation of new ideas ( Benedek et al., 2014 ) and together with the PPC in coupling different information together ( Simone Sandkühler, 2008 ; Stocco et al., 2012 ). Beaty et al. (2016) have proposed that even this internal idea-generation process can be goal directed, thereby allowing for a closer working relationship between the CEN and the DMN. They point to neuroimaging evidence that support the possibility that the executive control network (comprising the lateral prefrontal and inferior parietal regions) can constrain and direct the DMN in its process of generating ideas to meet task-specific goals via top down monitoring and executive control ( Beaty et al., 2016 ). The control network is believed to maintain an “internal train of thought” by keeping the task goal activated, thereby allowing for strategic and goal-congruent searches for ideas. Moreover, they suggest that the extent of CEN involvement in the DMN idea-generation may depend on the extent to which the creative task is constrained. In the RWPS setting, I would suspect that the internal search for creative solutions is not entirely unconstrained, even in the defocused mode. Instead, the solver is working on a specified problem and thus, must maintain the problem-thread while searching for solutions. Moreover, self-generated ideas must be evaluated against the problem parameters and thereby might need some top-down processing. This would suggest that in such circumstances, we would expect to see an increased involvement of the CEN in constraining the DMN.

On the external front, several mechanisms are operating in this defocused mode. Of particular note are the dorsal attention network, composed of the visual cortex (V), IPS and the frontal eye field (FEF) along with the precuneus and the caudate nucleus allow for partial cues to be considered. The MTL receives synthesized cue and contextual information and populates the WM in the PFC with a potentially expanded set of information that might be relevant for problem-solving. The precuneus, dlPFC and PPC together trigger the activation and use of a heuristic prototype based on an event in the environment. The caudate nucleus facilitates information routing between the PFC and PPC and is involved in learning and skill acquisition.

5.2.4. Focusing Event-Triggered Mode Switching

The problem's life in this defocused mode continues until a focusing event occurs, which could be triggered by either external (e.g., notification of impending deadline, discovery of a novel property in the environment) or internal items (e.g., goal completion, discovery of novel association or updated relevancy of a previously irrelevant item). As noted earlier, an internal train of thought may be maintained that facilitates top-down evaluation of ideas and tracking of these triggers ( Beaty et al., 2016 ). The salience network switches various networks back to the focused problem-solving mode, but not without the potential for problem restructuring. As noted earlier, problem space elements are maintained somewhat loosely in the defocused mode. Thus, upon a focusing event, a set or subset of these elements cohere into a tight (restructured) representation suitable for focused mode problem solving. The process then repeats itself until the goal has been achieved.

5.3. Model Predictions

5.3.1. single-mode operation.

The proposed RWPS model provides several interesting hypotheses, which I discuss next. First, the model assumes that any given problem being worked on is in one mode or another, but not both. Thus, the model predicts that there cannot be focused plan execution on a problem that is in defocused mode. The corollary prediction is that novel perceptual cues (as those discussed in section 4) cannot help the solver when in focused mode. The corollary prediction, presumably has some support from the inattentional blindness literature. Inattentional blindness is when perceptual cues are not noticed during a task (e.g., counting the number of basketball passes between several people, but not noticing a gorilla in the scene) ( Simons and Chabris, 1999 ). It is possible that during focused problem solving, that external and internally generated novel ideas are simply not considered for problem solving. I am not claiming that these perceptual cues are always ignored, but that they are not considered within the problem. Sometimes external cues (like distracting occurrences) can serve as defocusing events, but the model predicts that the actual content of these cues are not themselves useful for solving the specific problem at hand.

When comparing dual-process models Sowden et al. (2015) discuss shifting from one type of thinking to another and explore how this shift relates to creativity. In this regard, they weigh the pros and cons of serial vs. parallel shifts. In dual-process models that suggest serial shifts, it is necessary to disengage one type of thought prior to engaging the other or to shift along a continuum. Whereas, in models that suggest parallel shifts, each of the thinking types can operate in parallel. Per this construction, the proposed RWPS model is serial, however, not quite in the same sense. As noted earlier, the RWPS model is not a dual-process model in the same sense as other dual process model. Instead, here, the thrust is on when the brain is receptive or not receptive to certain kinds of internal and external stimuli that can influence problem solving. Thus, while the modes may be serial with respect to a certain problem, it does not preclude the possibility of serial and parallel thinking processes that might be involved within these modes.

5.3.2. Event-Driven Transitions

The model requires an event (defocusing or focusing) to transition from one mode to another. After all why else would a problem that is successfully being resolved in the focused mode (toward completion) need to necessarily be transferred to defocused mode? These events are interpreted as conflicts in the brain and therefore the mode-switching is enabled by the saliency network and the ACC. Thus, the model predicts that there can be no transition from one mode to another without an event. This is a bit circular, as an event is really what triggers the transition in the first place. But, here I am suggesting that an external or internal cue triggered event is what drives the transition, and that transitions cannot happen organically without such an event. In some sense, the argument is that the transition is discontinuous, rather than a smooth one. Mind-wandering is good example of when we might drift into defocused mode, which I suggest is an example of an internally driven event caused by an alternative thought that takes attention away from the problem.

A model assumption underlying RWPS is that events such as impasses have a similar effect to other events such as distraction or mind wandering. Thus, it is crucial to be able to establish that there exists of class of such events and they have a shared effect on RWPS, which is to switch attentional modes.

5.3.3. Focused Mode Completion

The model also predicts that problems cannot be solved (i.e., completed) within the defocused mode. A problem can be considered solved when a goal is reached. However, if a goal is reached and a problem is completed in the defocused mode, then there must have not been any converging event or coherence of problem elements. While it is possible that the solver arbitrarily arrived at the goal in a diffused problem space and without conscious awareness of completing the task or even any converging event or problem recompiling, it appears somewhat unlikely. It is true that there are many tasks that we complete without actively thinking about it. We do not think about what foot to place in front of another while walking, but this is not an instance of problem solving. Instead, this is an instance of unconscious task completion.

5.3.4. Restructuring Required

The model predicts that a problem cannot return to a focused mode without some amount of restructuring. That is, once defocused, the problem is essentially never the same again. The problem elements begin interacting with other internally and externally-generated items, which in turn become absorbed into the problem representation. This prediction can potentially be tested by establishing some preliminary knowledge, and then showing one group of subjects the same knowledge as before, while showing the another group of subjects different stimuli. If the model's predictions hold, the problem representation will be restructured in some way for both groups.

There are numerous other such predictions, which are beyond the scope of this paper. One of the biggest challenges then becomes evaluating the model to set up suitable experiments aimed at testing the predictions and falsifying the theory, which I address next.

6. Experimental Challenges and Paradigms

One of challenges in evaluating the RWPS is that real world factors cannot realistically be accounted for and sufficiently controlled within a laboratory environment. So, how can one controllably test the various predictions and model assumptions of “real world” problem solving, especially given that by definition RWPS involves the external environment and unconscious processing? At the expense of ecological validity, much of insight problem solving research has employed an experimental paradigm that involves providing participants single instances of suitably difficult problems as stimuli and observing various physiological, neurological and behavioral measures. In addition, through verbal protocols, experimenters have been able to capture subjective accounts and problem solving processes that are available to the participants' conscious. These experiments have been made more sophisticated through the use of timed-hints and/or distractions. One challenge with this paradigm has been the selection of a suitable set of appropriately difficult problems. The classic insight problems (e.g., Nine-dot, eight-coin) can be quite difficult, requiring complicated problem solving processes, and also might not generalize to other problems or real world problems. Some in the insight research community have moved in the direction of verbal tasks (e.g., riddles, anagrams, matchstick rebus, remote associates tasks, and compound remote associates tasks). Unfortunately, these puzzles, while providing a great degree of controllability and repeatability, are even less realistic. These problems are not entirely congruent with the kinds of problems that humans are solving every day.

The other challenge with insight experiments is the selection of appropriate performance and process tracking measures. Most commonly, insight researchers use measures such as time to solution, probability of finding solution, and the like for performance measures. For process tracking, verbal protocols, coded solution attempts, and eye tracking are increasingly common. In neuroscientific studies of insight various neurological measures using functional magnetic resonance imaging (fMRI), electroencephalography (EEGs), transcranial direct current stimulation (tDCS), and transcranial magnetic stimulation (tMS) are popular and allow for spatially and temporally localizing an insight event.

Thus, the challenge for RWPS is two-fold: (1) selection of stimuli (real world problems) that are generalizable, and (2) selection of measures (or a set of measures) that can capture key aspects of the problem solving process. Unfortunately, these two challenges are somewhat at odds with each other. While fMRI and various neuroscientific measures can capture the problem solving process in real time, it is practically difficult to provide participants a realistic scenario while they are laying flat on their back in an fMRI machine and allowed to move nothing more than a finger. To begin addressing this conundrum, I suggest returning to object manipulation problems (not all that different from those originally introduced by Maier and Duncker nearly a century ago), but using modern computing and user-interface technologies.

One pseudo-realistic approach is to generate challenging object manipulation problems in Virtual Reality (VR). VR has been used to describe 3-D environment displays that allows participants to interact with artificially projected, but experientially realistic scenarios. It has been suggested that virtual environments (VE) invoke the same cognitive modules as real equivalent environmental experience ( Foreman, 2010 ). Crucially, since VE's can be scaled and designed as desired, they provide a unique opportunity to study pseudo-RWPS. However, a VR-based research approach has its limitations, one of which is that it is nearly impossible to track participant progress through a virtual problem using popular neuroscientific measures such as fMRI because of the limited mobility of connected participants.

Most of the studies cited in this paper utilized an fMRI-based approach in conjunction with a verbal or visual task involving problem-solving or creative thinking. Very few, if any, studies involved the use physical manipulation, and those physical manipulations were restricted to limited finger movements. Thus, another pseudo-realistic approach is allowing subjects to teleoperate robotic arms and legs from inside the fMRI machine. This paradigm has seen limited usage in psychology and robotics, in studies focused on Human-Robot interaction ( Loth et al., 2015 ). It could be an invaluable tool in studying real-time dynamic problem-solving through the control of a robotic arm. In this paradigm a problem solving task involving physical manipulation is presented to the subject via the cameras of a robot. The subject (in an fMRI) can push buttons to operate the robot and interact with its environment. While the subjects are not themselves moving, they can still manipulate objects in the real world. What makes this paradigm all the more interesting is that the subject's manipulation-capabilities can be systematically controlled. Thus, for a particular problem, different robotic perceptual and manipulation capabilities can be exposed, allowing researchers to study solver-problem dynamics in a new way. For example, even simple manipulation problems (e.g., re-arranging and stacking blocks on a table) can be turned into challenging problems when the robotic movements are restricted. Here, the problem space restrictions are imposed not necessarily on the underlying problem, but on the solver's own capabilities. Problems of this nature, given their simple structure, may enable studying everyday practical creativity without the burden of devising complex creative puzzles. Crucial to note, both these pseudo-realistic paradigms proposed demonstrate a tight interplay between the solver's own capabilities and their environment.

7. Conclusion

While the neural basis for problem-solving, creativity and insight have been studied extensively in the past, there is still a lack of understanding of the role of the environment in informing the problem-solving process. Current research has primarily focused on internally-guided mental processes for idea generation and evaluation. However, the type of real world problem-solving (RWPS) that is often considered a hallmark of human intelligence has involved both a dynamic interaction with the environment and the ability to handle intervening and interrupting events. In this paper, I have attempted to synthesize the literature into a unified theory of RWPS, with a specific focus on ways in which the environment can help problem-solve and the key neural networks involved in processing and utilizing relevant and useful environmental information. Understanding the neural basis for RWPS will allow us to be better situated to solve difficult problems. Moreover, for researchers in computer science and artificial intelligence, clues into the neural underpinnings of the computations taking place during creative RWPS, can inform the design the next generation of helper and exploration robots which need these capabilities in order to be resourceful and resilient in the open-world.

Author Contributions

The author confirms being the sole contributor of this work and approved it for publication.

The research for this Hypothesis/Theory Article was funded by the authors private means. Publication costs will be covered by my institution: Tufts University, Medford, MA, USA.

Conflict of Interest Statement

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

I am indebted to Professor Matthias Scheutz, Professor Elizabeth Race, Professor Ayanna Thomas, and Professor. Shaun Patel for providing guidance with the research and the manuscript. I am also grateful for the facilities provided by Tufts University, Medford, MA, USA.

1. ^ My intention is not to ignore the benefits of a concentrated internal thought process which likely occurred as well, but merely to acknowledge the possibility that the environment might have also helped.

2. ^ The research in insight does extensively use “hints” which are, arguably, a form of external influence. But these hints are highly targeted and might not be available in this explicit form when solving problems in the real world.

3. ^ The accuracy of these accounts has been placed in doubt. They often are recounted years later, with inaccuracies, and embellished for dramatic effect.

4. ^ I use the term “agent” to refer to the problem-solver. The term agent is more general than “creature” or “person” or “you" and is intentionally selected to broadly reference humans, animals as well as artificial agents. I also selectively use the term “solver.”

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Keywords: creativity, problem-solving, insight, attention network, salience network, default mode network

Citation: Sarathy V (2018) Real World Problem-Solving. Front. Hum. Neurosci . 12:261. doi: 10.3389/fnhum.2018.00261

Received: 03 August 2017; Accepted: 06 June 2018; Published: 26 June 2018.

Reviewed by:

Copyright © 2018 Sarathy. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Vasanth Sarathy, [email protected]

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October 1, 2018

To Solve Real-World Problems, We Need Interdisciplinary Science

Solving today’s complex, global problems will take interdisciplinary science

By Graham A. J. Worthy & Cherie L. Yestrebsky

T he Indian River Lagoon, a shallow estuary that stretches for 156 miles along Florida's eastern coast, is suffering from the activities of human society. Poor water quality and toxic algal blooms have resulted in fish kills, manatee and dolphin die-offs, and takeovers by invasive species. But the humans who live here have needs, too: the eastern side of the lagoon is buffered by a stretch of barrier islands that are critical to Florida's economy, tourism and agriculture, as well as for launching NASA missions into space.

As in Florida, many of the world's coastlines are in serious trouble as a result of population growth and the pollution it produces. Moreover, the effects of climate change are accelerating both environmental and economic decline. Given what is at risk, scientists like us—a biologist and a chemist at the University of Central Florida—feel an urgent need to do research that can inform policy that will increase the resiliency and sustainability of coastal communities. How can our research best help balance environmental and social needs within the confines of our political and economic systems? This is the level of complexity that scientists must enter into instead of shying away from.

Although new technologies will surely play a role in tackling issues such as climate change, rising seas and coastal flooding, we cannot rely on innovation alone. Technology generally does not take into consideration the complex interactions between people and the environment. That is why coming up with solutions will require scientists to engage in an interdisciplinary team approach—something that is common in the business world but is relatively rare in universities.

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Universities are a tremendous source of intellectual power, of course. But students and faculty are typically organized within departments, or academic silos. Scientists are trained in the tools and language of their respective disciplines and learn to communicate their findings to one another using specific jargon.

When the goal of research is a fundamental understanding of a physical or biological system within a niche community, this setup makes a lot of sense. But when the problem the research is trying to solve extends beyond a closed system and includes its effects on society, silos create a variety of barriers. They can limit creativity, flexibility and nimbleness and actually discourage scientists from working across disciplines. As professors, we tend to train our students in our own image, inadvertently producing specialists who have difficulty communicating with the scientist in the next building—let alone with the broader public. This makes research silos ineffective at responding to developing issues in policy and planning, such as how coastal communities and ecosystems worldwide will adapt to rising seas.

Science for the Bigger Picture

As scientists who live and work in Florida, we realized that we needed to play a bigger role in helping our state—and country—make evidence-based choices when it comes to vulnerable coastlines. We wanted to make a more comprehensive assessment of both natural and human-related impacts to the health, restoration and sustainability of our coastal systems and to conduct long-term, integrated research.

At first, we focused on expanding research capacity in our biology, chemistry and engineering programs because each already had a strong coastal research presence. Then, our university announced a Faculty Cluster Initiative, with a goal of developing interdisciplinary academic teams focused on solving tomorrow's most challenging societal problems. While putting together our proposal, we discovered that there were already 35 faculty members on the Orlando campus who studied coastal issues. They belonged to 12 departments in seven colleges, and many of them had never even met. It became clear that simply working on the same campus was insufficient for collaboration.

So we set out to build a team of people from a wide mix of backgrounds who would work in close proximity to one another on a daily basis. These core members would also serve as a link to the disciplinary strengths of their tenure home departments. Initially, finding experts who truly wanted to embrace the team aspect was more difficult than we thought. Although the notion of interdisciplinary research is not new, it has not always been encouraged in academia. Some faculty who go in that direction still worry about whether it will threaten their recognition when applying for grants, seeking promotions or submitting papers to high-impact journals. We are not suggesting that traditional academic departments should be disbanded. On the contrary, they give the required depth to the research, whereas the interdisciplinary team gives breadth to the overall effort.

Our cluster proposal was a success, and in 2019 the National Center for Integrated Coastal Research (UCF Coastal) was born. Our goal is to guide more effective economic development, environmental stewardship, hazard-mitigation planning and public policy for coastal communities. To better integrate science with societal needs, we have brought together biologists, chemists, engineers and biomedical researchers with anthropologists, sociologists, political scientists, planners, emergency managers and economists. It seems that the most creative perspectives on old problems have arisen when people with different training and life experiences are talking through issues over cups of coffee. After all, "interdisciplinary" must mean more than just different flavors of STEM. In academia, tackling the effects of climate change demands more rigorous inclusion of the social sciences—an area that has been frequently overlooked.

The National Science Foundation, as well as other groups, requires that all research proposals incorporate a social sciences component, as an attempt to assess the broader implications of projects. Unfortunately, in many cases, a social scientist is simply added to a proposal only to check a box rather than to make a true commitment to allowing that discipline to inform the project. Instead social, economic and policy needs must be considered at the outset of research design, not as an afterthought. Otherwise our work might fail at the implementation stage, which means we will not be as effective as we could be in solving real-world problems. As a result, the public might become skeptical about how much scientists can contribute toward solutions.

Connecting with the Public

The reality is that communicating research findings to the public is an increasingly critical responsibility of scientists. Doing so has a measurable effect on how politicians prioritize policy, funding and regulations. UCF Coastal was brought into a world where science is not always respected—sometimes it is even portrayed as the enemy. There has been a significant erosion of trust in science over recent years, and we must work more deliberately to regain it. The public, we have found, wants to see quality academic research that is grounded in the societal challenges we are facing. That is why we are melding pure academic research with applied research to focus on issues that are immediate—helping a town or business recovering from a hurricane, for example—as well as long term, such as directly advising a community on how to build resiliency as flooding becomes more frequent.

As scientists, we cannot expect to explain the implications of our research to the wider public if we cannot first understand one another. A benefit of regularly working side by side is that we are crafting a common language, reconciling the radically different meanings that the same words can have to a variety of specialists. Finally, we are learning to speak to one another with more clarity and understand more explicitly how our niches fit into the bigger picture. We are also more aware of culture and industry as driving forces in shaping consensus and policy. Rather than handing city planners a stack of research papers and walking away, UCF Coastal sees itself as a collaborator that listens instead of just lecturing.

This style of academic mission is not only relevant to issues around climate change. It relates to every aspect of modern society, including genetic engineering, automation, artificial intelligence, and so on. The launch of UCF Coastal garnered positive attention from industry, government agencies, local communities and academics. We think that is because people do want to come together to solve problems, but they need a better mechanism for doing so. We hope to be that conduit while inspiring other academic institutions to do the same.

After all, we have been told for years to "think globally, act locally" and that "all politics is local." Florida's Indian River Lagoon will be restored only if there is engagement among residents, local industries, academics, government agencies and nonprofit organizations. As scientists, it is our responsibility to help everyone involved understand that problems that took decades to create will take decades to fix. We need to present the most helpful solutions while explaining the intricacies of the trade-offs for each one. Doing so is possible only if we see ourselves as part of an interdisciplinary, whole-community approach. By listening and responding to fears and concerns, we can make a stronger case for why scientifically driven decisions will be more effective in the long run.

Graham A. J. Worthy is founder and director of the National Center for Integrated Coastal Research at the University of Central Florida (UCF Coastal) and chairs the university's department of biology. His research focuses on how marine ecosystems respond to natural and anthropogenic perturbations.

Cherie L. Yestrebsky is a professor in the University of Central Florida's department of chemistry. Her research expertise is in environmental chemistry and remediation of pollutants in the environment.

• Our Mission

Using Mathematical Modeling to Get Real With Students

Unlike canned word problems, mathematical modeling plunges students into the messy complexities of real-world problem solving.  

How do you bring math to life for kids? Illustrating the boundless possibilities of mathematics can be difficult if students are only asked to examine hypothetical situations like divvying up a dessert equally or determining how many apples are left after sharing with friends, writes third- and fourth- grade teacher Matthew Kandel for Mathematics Teacher: Learning and Teaching PK-12 .

In the early years of instruction, it’s not uncommon for students to think they’re learning math for the sole purpose of being able to solve word problems or help fictional characters troubleshoot issues in their imaginary lives, Kandel says. “A word problem is a one-dimensional world,” he writes. “Everything is distilled down to the quantities of interest. To solve a word problem, students can pick out the numbers and decide on an operation.” 

But through the use of mathematical modeling, students are plucked out of the hypothetical realm and plunged into the complexities of reality—presented with opportunities to help solve real-world problems with many variables by generating questions, making assumptions, learning and applying new skills, and ultimately arriving at an answer.

In Kandel’s classroom, this work begins with breaking students into small groups, providing them with an unsharpened pencil and a simple, guiding question: “How many times can a pencil be sharpened before it is too small to use?”

Setting the Stage for Inquiry 

The process of tackling the pencil question is not unlike the scientific method. After defining a question to investigate, students begin to wonder and hypothesize—what information do we need to know?—in order to identify a course of action. This step is unique to mathematical modeling: Whereas a word problem is formulaic, leading students down a pre-existing path toward a solution, a modeling task is “free-range,” empowering students to use their individual perspectives to guide them as they progress through their investigation, Kandel says. 

Modeling problems also have a number of variables, and students themselves have the agency to determine what to ignore and what to focus their attention on. 

After inter-group discussions, students in Kandel’s classroom came to the conclusion that they’d need answers to a host of other questions to proceed with answering their initial inquiry: 

• How much does the pencil sharpener remove? 
• What is the length of a brand new, unsharpened pencil? 
• Does the pencil sharpener remove the same amount of pencil each time it is used?

Introducing New Skills in Context

Once students have determined the first mathematical question they’d like to tackle (does the pencil sharpener remove the same amount of pencil each time it is used?), they are met with a roadblock. How were they to measure the pencil if the length did not fall conveniently on an inch or half inch? Kandel took the opportunity to introduce a new target skill which the class could begin using immediately: measuring to the nearest quarter inch. 

“One group of students was not satisfied with the precision of measuring to the nearest quarter inch and asked to learn how to measure to the nearest eighth of an inch,” Kandel explains. “The attention and motivation exhibited by students is unrivaled by the traditional class in which the skill comes first, the problem second.” 

Students reached a consensus and settled on taking six measurements total: the initial length of the new, unsharpened pencil as well as the lengths of the pencil after each of five sharpenings. To ensure all students can practice their newly acquired skill, Kandel tells the class that “all group members must share responsibility, taking turns measuring and checking the measurements of others.” 

Next, each group created a simple chart to record their measurements, then plotted their data as a line graph—though exploring other data visualization techniques or engaging students in alternative followup activities would work as well.

“We paused for a quick lesson on the number line and the introduction of a new term—mixed numbers,” Kandel explains. “Armed with this new information, students had no trouble marking their y-axis in half- or quarter-inch increments.” 

Sparking Mathematical Discussions

Mathematical modeling presents a multitude of opportunities for class-wide or small-group discussions, some which evolve into debates in which students state their hypotheses, then subsequently continue working to confirm or refute them. 

Kandel’s students, for example, had a wide range of opinions when it came to answering the question of how small of a pencil would be deemed unusable. Eventually, the class agreed that once a pencil reached 1 ¼ inch, it could no longer be sharpened—though some students said they would be able to still write with it. 

“This discussion helped us better understand what it means to make an assumption and how our assumptions affected our mathematical outcomes,” Kandel writes. Students then indicated the minimum size with a horizontal line across their respective graphs. 

Many students independently recognized the final step of extending their line while looking at their graphs. With each of the six points representing their measurements, the points descended downward toward the newly added horizontal “line of inoperability.” 

With mathematical modeling, Kandel says, there are no right answers, only models that are “more or less closely aligned with real-world observations.” Each group of students may come to a different conclusion, which can lead to a larger class discussion about accuracy. To prove their group had the most accurate conclusion, students needed to compare and contrast their methods as well as defend their final result. 

Developing Your Own Mathematical Models

The pencil problem is a great starting point for introducing mathematical modeling and free-range problem solving to your students, but you can customize based on what you have available and the particular needs of each group of students.

Depending on the type of pencil sharpener you have, for example, students can determine what constitutes a “fair test” and set the terms of their own inquiry. 

Additionally, Kandel suggests putting scaffolds in place to allow students who are struggling with certain elements to participate: Simplified rulers can be provided for students who need accommodations; charts can be provided for students who struggle with data collection; graphs with prelabeled x- and y-axes can be prepared in advance.

.css-1sk4066:hover{background:#d1ecfa;} 7 Real-World Math Strategies

Students can also explore completely different free-range problem solving and real world applications for math . At North Agincourt Jr. Public School in Scarborough, Canada, kids in grades 1-6 learn to conduct water audits. By adding, subtracting, finding averages, and measuring liquids—like the flow rate of all the water foundations, toilets, and urinals—students measure the amount of water used in their school or home in a single day. 

Or you can ask older students to bring in common household items—anything from a measuring cup to a recipe card—and identify three ways the item relates to math. At Woodrow Petty Elementary School in Taft, Texas, fifth-grade students display their chosen objects on the class’s “real-world math wall.” Even acting out restaurant scenarios can provide students with an opportunity to reinforce critical mathematical skills like addition and subtraction, while bolstering an understanding of decimals and percentages. At Suzhou Singapore International School in China, third- to fifth- graders role play with menus, ordering fictional meals and learning how to split the check when the bill arrives. 

Solving real-world problem using data science

Naman Doshi

Towards Data Science

The world of data science is evolving every day. Every professional in this field needs to be updated and constantly learning, or risk being left behind. You must have an appetite to solve problems. So I decided to study and solve a real-world problem which most of us have faced in our professional careers. The technical round in an interview!

How many times have you gone through a technical interview where you feel you’re acing it, and then a question comes that leaves you stumped? And from there the entire interview goes downhill because now you have lost confidence and the recruiter has lost interest.

But is it fair to judge the technical capabilities of a candidate based entirely on a 3-hour interview? This is a loss at both the ends because now the company has lost a potential candidate and the candidate has lost an opportunity.

If only there was a way through which the recruiter can get the gist about the technical capabilities of the candidate outside the interview hall. A scoring system of sorts — that would give an ideal score to gauge the technical knowledge of the candidate, and thereby help the recruiter to make an informed, unbiased decision. Sounds like a dream scenario, right?

So I decided to start a project called “Scorey” that aims to crack this challenge.

Scorey helps in scraping, aggregating, and assessing the technical ability of a candidate based on publicly available sources.

Setting the Problem Statement

The current interview scenario is biased towards “candidate’s performance during the 3-hour interview” and doesn’t take other factors into account, such as the candidate’s competitive coding abilities, contribution towards the developer community, and so on.

The Approach We’ll Take

Scorey tries to solve this problem by aggregating publicly available data from various websites, such as:

• StackOverflow
• Hackerearth

Once the data is collected, the algorithm then defines a comprehensive scoring system that grades the candidates technical capabilities based on the following factors:

• Number of Problems Solved
• Contribution

The candidate is then assigned a scored out of 100. This helps the interviewer get a full view of a candidate’s abilities and hence make an unbiased, informed decision.

Setting up the Project

For the entire scope of this project, we are going to use Python, a Jupyter notebook & scraping libraries. So if you’re someone who likes Notebooks, then this section is for you. If not, feel free to skip this and move on to the next section.

This will make your dataframe output look neat, tidy, and really good!

Now that we have a gist of what we are aiming to solve and how we are going to go about it, let’s code!

Step 1: Scraping Personal Website

We need to aggregate the entire “ coding presence of a person on the internet ”. But where do we start? Duh! His/Her personal website. This is of course assuming we have access and permission to the candidate’s personal website. We can parse all the necessary links from there.

When we run this piece of code, we get the below output:

Here we are using BeautifulSoup which is a popular scraping library. Using this block of code, we have direct links to a candidate’s online profile.

Now where will you begin if you had to assess a coder at a much more granular level?

So first, let us use Github API to get all the info we need of a particular user.

For our use case, we only need email, number of repositories, followers, hireable (true or false), current company and last recorded activity.

2. StackOverflow

Ah. Devs might not believe in God but StackOverflow is definitely a temple for them. As you may already know, it is very difficult to get a reputation on StackOverflow. For this step, you can use StackExchange’s API — it gives you user data such as reputation, no. of answers, etc.

We’ll then add these new attributes to our existing dataframe.

Now we are going to target and scrape global competitive programming platforms such as CodeChef, SPOJ, Codebuddy, Hackerearth, CodeForces & GitAwards (for a deeper insight into their projects).

All this scraping gave us a LOT of info as you can see. Code is pretty self explanatory. I’ve also documented using comments so that its easy to understand.

Without going into the nitty-gritty of the code, I’d like to focus on the process. But you can give me a shout-out if you face any trouble executing the code. :) Now that we have all the data in hand, we will move on to creating a scoring algorithm.

Step 2: Scoring System

The next part is to score the candidates on the following parameters:

• Rank (25 points)
• Number of problems solved (25 points)
• Reputation (25 points)
• Followers (15 points)
• Activity (5 points)
• Contributions (5 points)

So if you go through this piece of code, you’d understand how we can create a scoring system. Though its pretty basic at this point, we can use machine learning to create a robust dynamic scoring system.

Final Score

Based on the point system we saw above, the algorithm will now assign a final score to the candidate’s technical capabilities.

So the user poke19962008 has a score of 64 out of 100 ! Now this will give the recruiters an idea of the technical abilities of the candidate outside the interview room.

Step 3: Predictive Modeling

When you are trying to solve a real world problem and “ productivize ” the solution, its important to consider the requirements of the end user. In this case, its the recruiter. How can we use power of machine learning to add value to the recruiter?

Upon brainstorming, I found the following use cases — 1. Model that predicts whether or not the management will be satisfied by candidate’s skill set 2. Model that predicts the probability of a candidate’s churn post hiring 3. Using genetic algorithm to link assign the candidate to respective team

Let’s try to code the 1st use case — Predicting company’s satisfaction Assuming that the recruiter has been using Scorey to screen candidates for some time and now has a database of 100 candidates. Post recruitment, based on the candidate’s performance, the recruiter updates the database with a new binary attribute “Satisfaction” with values of either 0 or 1. Let’s create a dummy database for now and try to create a model using Scikit-Learn, Pandas, Numpy and build a predictive model.

• Import data & libraries
• Clean the data — remove duplicates and null values
• Using label encoder to deal with categorical data
• Split the dataset into train & test
• Using kNN classifier to predict
• Check Accuracy

Using these steps, you will get a niche model that will be able to predict whether the candidate will fit into the company based on underlying trends. For eg — Candidates who have higher reputation and are contributing to Open source are more likely to retain for a longer period of time.

Step 4: Dashboarding

I went ahead and made a dashboard. This its still a work-in-progress and I’ll be happy to share some of the screenshots of the interface.

There you have it. Your very own end-to-end product. To summarize — 1. We identified a problem 2. Methodical thinking on how we can solve it 3. Used Web scraping to gather data 4. Build an algorithmic scoring system 5. Machine learning to build a predictive model 5. Dashboard to communicate results

Tech stack that we used — Python: BeautifulSoup, Urllib, Pandas, Sklearn

So that’s all for this article. We took a real life problem and tried to use data and algorithms to solve it!

Next time you go for an interview, you can pitch this system to the recruiter. :)

So what’s next?

Code for the entire project can be found on Github — here

• Integrating Machine learning components for rule generation
• Handling missing data exceptions dynamically

If you think this project is cool and would like to contribute, you are more than welcome! Let’s build something exciting for the community.

You can connect with me over LinkedIn or on Twitter to get daily updates on what’s new in data science & machine learning.

Written by Naman Doshi

Writes about data science, payments, ecommerce, behavioral economics

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Problem Solving Using Computational Thinking

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Instructor: Chris Quintana

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What you'll learn

Recognize Computational Thinking concepts in practice through a series of real-world case examples.

Develop solutions through the application of Computational Thinking concepts to real world problems.

Skills you'll gain

• Computer Programming
• Computational Thinking

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There are 5 modules in this course

Have you ever heard that computers "think"? Believe it or not, computers really do not think. Instead, they do exactly what we tell them to do. Programming is, "telling the computer what to do and how to do it."

Before you can think about programming a computer, you need to work out exactly what it is you want to tell the computer to do. Thinking through problems this way is Computational Thinking. Computational Thinking allows us to take complex problems, understand what the problem is, and develop solutions. We can present these solutions in a way that both computers and people can understand. The course includes an introduction to computational thinking and a broad definition of each concept, a series of real-world cases that illustrate how computational thinking can be used to solve complex problems, and a student project that asks you to apply what they are learning about Computational Thinking in a real-world situation. This project will be completed in stages (and milestones) and will also include a final disaster response plan you'll share with other learners like you. This course is designed for anyone who is just beginning programming, is thinking about programming or simply wants to understand a new way of thinking about problems critically. No prior programming is needed. The examples in this course may feel particularly relevant to a High School audience and were designed to be understandable by anyone. You will learn: -To define Computational Thinking components including abstraction, problem identification, decomposition, pattern recognition, algorithms, and evaluating solutions -To recognize Computational Thinking concepts in practice through a series of real-world case examples -To develop solutions through the application of Computational Thinking concepts to real world problems

Foundations of Computational Thinking

What's included.

3 videos 5 readings 2 quizzes 1 discussion prompt

3 videos • Total 43 minutes

• Welcome to Computational Thinking • 15 minutes • Preview module
• Example: Making a Cake • 16 minutes
• Introduction to the Graphic Organizer • 11 minutes

5 readings • Total 50 minutes

• Welcome and Syllabus • 10 minutes
• Help Us Learn More about You! • 10 minutes
• Contributor Acknowledgements • 10 minutes
• Introduction to the Graphic Organizer • 10 minutes
• Would you like to plan your learning journey with Michigan Online? • 10 minutes

2 quizzes • Total 35 minutes

• Foundations of Computational Thinking Quiz • 25 minutes
• Foundations of Computational Thinking Practice Questions • 10 minutes

1 discussion prompt • Total 10 minutes

• Real-World Applications of Computational Thinking • 10 minutes

Case Study: Airport Surveillance and Image Analysis

6 videos 3 readings 3 quizzes 2 discussion prompts

6 videos • Total 29 minutes

• Image Analysis: Importance of Computational Thinking - Part 1 • 2 minutes • Preview module
• Image Analysis: Importance of Computational Thinking - Part 2 • 1 minute
• Image Analysis: Abstraction and Algorithms • 10 minutes
• Image Analysis: Algorithms, Optional Advanced Video • 9 minutes
• Image Analysis: Evaluating Solutions • 4 minutes
• Image Analysis: Problem Identification and Decomposition • 0 minutes

3 readings • Total 30 minutes

• Introduction to Airport Surveillance Case-Study • 10 minutes
• Airport Surveillance Case-Study Check-In 1 • 10 minutes
• Airport Surveillance Check-In 2 • 10 minutes

3 quizzes • Total 55 minutes

• Airport Surveillance Case-Study Quiz • 20 minutes
• Airport Surveillance Practice Questions Set 1 • 15 minutes
• Airport Surveillance Practice Questions Set 2 • 20 minutes

2 discussion prompts • Total 20 minutes

• Image Analysis: What Would You Do? • 10 minutes
• Other Applications • 10 minutes

Case Study: Epidemiology

6 videos 5 readings 2 quizzes 2 discussion prompts

6 videos • Total 52 minutes

• Epidemiology: Introduction and Problem Identification • 1 minute • Preview module
• Epidemiology: Problem Identification Part 2 • 8 minutes
• Epidemiology: Abstraction and Decomposition • 14 minutes
• Epidemiology: Algorithms and Evaluating Solutions - Part 1 • 13 minutes
• Epidemiology: Algorithms and Evaluating Solutions - Part 2 • 8 minutes
• Epidemiology: Conclusion • 6 minutes
• Introduction to Epidemiology Case-Study • 10 minutes
• Epidemiology Case-Study Check-In 1 • 10 minutes
• Up Next: Rafael's Algorithm • 10 minutes
• Epidemiology Case-Study Check-In 2 • 10 minutes
• Stay in touch on University of Michigan online courses • 10 minutes

2 quizzes • Total 36 minutes

• Epidemiology Case-Study Quiz • 20 minutes
• Epidemiology Practice Questions • 16 minutes
• Using Computational Thinking in Public Health • 10 minutes
• Understanding the Problem • 10 minutes

Case Study: Human Trafficking

3 videos 2 readings 2 quizzes

3 videos • Total 34 minutes

• Human Trafficking: Importance of Computational Thinking • 10 minutes • Preview module
• Human Trafficking: How Computational Thinking May Apply - Part 1 • 11 minutes
• Human Trafficking: How Computational Thinking May Apply - Part 2 • 12 minutes

2 readings • Total 20 minutes

• Introduction to Human Trafficking Case-Study • 10 minutes
• Human Trafficking Case-Study Check-In • 10 minutes

2 quizzes • Total 50 minutes

• Next Case: Potential Applications of Computational Thinking to Human Trafficking • 30 minutes
• Human Trafficking Practice Questions • 20 minutes

Final Project

8 readings 1 peer review

8 readings • Total 80 minutes

• Introduction to the Final Project • 10 minutes
• Final Project Part 1. Background and Context • 10 minutes
• Final Project Part 2: Graphic Organizer and Project Justification • 10 minutes
• Final Project Part 3: Project Justification • 10 minutes
• Final Project Part 4: Algorithm depiction • 10 minutes
• Course Feedback • 10 minutes
• Create innovative learning environments for students with Introduction to Learning Experience Design • 10 minutes
• Keep Learning with Michigan Online • 10 minutes

1 peer review • Total 60 minutes

• Final Project • 60 minutes

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Reviewed on May 27, 2021

The course helped me develop problem thinking skills and I appreciate the real life examples used in teaching the course. They made understanding the concepts much easier.

Reviewed on Jul 27, 2021

This course is what I really need to understand what is Computational Thinking. I learned about all aspect of it. To who want to begin your road to Computer Science, this course is my recommend

Reviewed on May 30, 2021

The course is highly enlightening. It has helped me see that a lot of problems can be solved using computational thinking. I will recommend to anyone willing to gain knowledge in this area.

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Multiple goals, multiple solutions, plenty of second-guessing and revising − here’s how science really works

Professor of Philosophy, University of Montana

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Soazig Le Bihan receives funding from the Maureen and Mike Mansfield Center at the University of Montana.

University of Montana provides funding as a member of The Conversation US.

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A man in a lab coat bends under a dim light, his strained eyes riveted onto a microscope. He’s powered only by caffeine and anticipation.

This solitary scientist will stay on task until he unveils the truth about the cause of the dangerous disease quickly spreading through his vulnerable city. Time is short, the stakes are high, and only he can save everyone. …

That kind of romanticized picture of science was standard for a long time. But it’s as far from actual scientific practice as a movie’s choreographed martial arts battle is from a real fistfight.

For most of the 20th century, philosophers of science like me maintained somewhat idealistic claims about what good science looks like. Over the past few decades, however, many of us have revised our views to better mirror actual scientific practice .

An update on what to expect from actual science is overdue. I often worry that when the public holds science to unrealistic standards, any scientific claim failing to live up to them arouses suspicion. While public trust is globally strong and has been for decades, it has been eroding. In November 2023, Americans’ trust in scientists was 14 points lower than it had been just prior to the COVID-19 pandemic, with its flurry of confusing and sometimes contradictory science-related messages.

When people’s expectations are not met about how science works, they may blame scientists. But modifying our expectations might be more useful. Here are three updates I think can help people better understand how science actually works. Hopefully, a better understanding of actual scientific practice will also shore up people’s trust in the process.

The many faces of scientific research

First, science is a complex endeavor involving multiple goals and associated activities.

Some scientists search for the causes underlying some observable effect, such as a decimated pine forest or the Earth’s global surface temperature increase .

Others may investigate the what rather than the why of things. For example, ecologists build models to estimate gray wolf abundance in Montana . Spotting predators is incredibly challenging. Counting all of them is impractical. Abundance models are neither complete nor 100% accurate – they offer estimates deemed good enough to set harvesting quotas. Perfect scientific models are just not in the cards .

Beyond the what and the why, scientists may focus on the how. For instance, the lives of people living with chronic illnesses can be improved by research on strategies for managing disease – to mitigate symptoms and improve function, even if the true causes of their disorders largely elude current medicine.

It’s understandable that some patients may grow frustrated or distrustful of medical providers unable to give clear answers about what causes their ailment. But it’s important to grasp that lots of scientific research focuses on how to effectively intervene in the world to reach some specific goals.

Simplistic views represent science as solely focused on providing causal explanations for the various phenomena we observe in this world. The truth is that scientists tackle all kinds of problems, which are best solved using different strategies and approaches and only sometimes involve full-fledged explanations.

Complex problems call for complex solutions

The second aspect of scientific practice worth underscoring is that, because scientists tackle complex problems, they don’t typically offer one unique, complete and perfect answer. Instead they consider multiple, partial and possibly conflicting solutions.

Scientific modeling strategies illustrate this point well. Scientific models typically are partial, simplified and sometimes deliberately unrealistic representations of a system of interest. Models can be physical, conceptual or mathematical. The critical point is that they represent target systems in ways that are useful in particular contexts of inquiry. Interestingly, considering multiple possible models is often the best strategy to tackle complex problems.

Scientists consider multiple models of biodiversity , atomic nuclei or climate change . Returning to wolf abundance estimates, multiple models can also fit the bill. Such models rely on various types of data, including acoustic surveys of wolf howls, genetic methods that use fecal samples from wolves, wolf sightings and photographic evidence, aerial surveys, snow track surveys and more.

Weighing the pros and cons of various possible solutions to the problem of interest is part and parcel of the scientific process. Interestingly, in some cases, using multiple conflicting models allows for better predictions than trying to unify all the models into one.

The public may be surprised and possibly suspicious when scientists push forward multiple models that rely on conflicting assumptions and make different predictions. People often think “real science” should provide definite, complete and foolproof answers to their questions. But given various limitations and the world’s complexity, keeping multiple perspectives in play is most often the best way for scientists to reach their goals and solve the problems at hand.

Science as a collective, contrarian endeavor

Finally, science is a collective endeavor, where healthy disagreement is a feature, not a bug.

The romanticized version of science pictures scientists working in isolation and establishing absolute truths. Instead, science is a social and contrarian process in which the community’s scrutiny ensures we have the best available knowledge. “Best available” does not mean “definitive,” but the best we have until we find out how to improve it. Science almost always allows for disagreements among experts.

Controversies are core to how science works at its best and are as old as Western science itself. In the 1600s, Descartes and Leibniz fought over how to best characterize the laws of dynamics and the nature of motion.

The long history of atomism provides a valuable perspective on how science is an intricate and winding process rather than a fast-delivery system of results set in stone. As Jean Baptiste Perrin conducted his 1908 experiments that seemingly settled all discussion regarding the existence of atoms and molecules, the questions of the atom’s properties were about to become the topic of decades of controversies with the birth of quantum physics.

The nature and structure of fundamental particles and associated fields have been the subject of scientific research for more than a century. Lively academic discussions abound concerning the difficult interpretation of quantum mechanics , the challenging unification of quantum physics and relativity , and the existence of the Higgs boson , among others.

Distrusting researchers for having healthy scientific disagreements is largely misguided.

A very human practice

To be clear, science is dysfunctional in some respects and contexts. Current institutions have incentives for counterproductive practices, including maximizing publication numbers . Like any human endeavor, science includes people with bad intent, including some trying to discredit legitimate scientific research . Finally, science is sometimes inappropriately influenced by various values in problematic ways.

These are all important considerations when evaluating the trustworthiness of particular scientific claims and recommendations. However, it is unfair, sometimes dangerous, to mistrust science for doing what it does at its best. Science is a multifaceted endeavor focused on solving complex problems that typically just don’t have simple solutions. Communities of experts scrutinize those solutions in hopes of providing the best available approach to tackling the problems of interest.

Science is also a fallible and collective process. Ignoring the realities of that process and holding science up to unrealistic standards may result in the public calling science out and losing trust in its reliability for the wrong reasons.

• Philosophy of science
• Scientific research
• Science myths
• Trust in science

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Global Nonprofit Technovation Helps Girls Solve Local Problems Using AI

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Earlier this year, nonprofit Technovation announced The AI Forward Alliance , a collaboration intended to increase the number of young women in the AI industry. This is one of many initiatives the organization has leveraged to keep its curriculum cutting-edge since its inception in 2006.

We spoke to founder Tara Chklovski, Ph.D., about Technovation’s mission, its AI initiatives, and how businesses can support young women in their careers. Bringing more women and girls into the AI industry brings the courage needed to solve problems like bias, she said.

What is Technovation?

Technovation is an educational nonprofit that operates around the world. Girls and young women ages 8 to 18 can join the program to create a project that benefits their local communities using technology. The project takes about 12 weeks.

Each girl or team of girls is paired with a mentor — for younger girls, a parent — who helps them identify problems and find technological solutions. One group from a rural area in Kenya trained AI to recognize gunshot sounds in order to speed up responses to crime from law enforcement. Other groups created apps to assist women in reporting domestic violence or an app connected to a vibrating bracelet that alerts deaf or hard-of-hearing people to fire, weather, or other alarms.

“We are the only program that is global, fully focused on girls, and has long-term data to show that because of the deep technology experience they go into higher degrees in computer science and into the tech careers at a much higher rate than normal,” said Chklovski.

She said 76% of Technovation graduates go into computer science degrees, and 60% go into tech careers, specifically because of their participation in the program. Chklovski attributes this success in part to putting problem-solving first.

“Instead of walking you through the fundamentals of programming and, at the end, you have a project, we flipped it to say, how do you identify meaningful problems that are going to change the world?” she said.

Participants emerge from their project with a robust business plan and pitch, as well as demo videos for their product.

Encouraging girls to get into forward-looking tech enriches the industry with people confident enough to introduce new ideas, Chklovski said.

“You just want a workforce that’s courageous, that can come up with new ideas,” she said. “The heart of innovation is new ideas, different ideas, different ways of thinking.”

AI is the latest of Technovation’s tools for solving local problems

The AI Forward Alliance is a collaboration between Technovation, UNICEF, Google, and other organizations with the goal of impacting 25 million young women. Specifically, the alliance seeks to foster problem-solving skills, complex systems thinking, data science, and machine learning.

Chklovski pointed out that AI isn’t Technovation’s sole focus, but it has been a part of its efforts for several years.

“We started doing an AI-in-action kind of curriculum eight years ago,” said Chklovski. “We are coming to this AI conversation with … tons of data on what works, what doesn’t work. The whole [generative] AI boom is only accelerating. What we have seen is exciting and interesting to young girls.

“The AI Forward Alliance really is about bringing cutting-edge technology. At the moment, it’s AI, but maybe in five to 10 years, maybe it’s quantum computing. So, we are in some sense tech-agnostic. It’s more about what’s relevant to workforce training, what’s the most powerful tool to tackle the big, complex problems we face.”

Bad data in, bad data out

Chklovski acknowledges that generative AI can cause, as The AI Forward Alliance stated, “contention and concern” — particularly when it comes to AI hallucinations, which occur when an AI model generates inaccurate or misleading information but presents it as if it were true.

“There’s so much room for improvement,” Chklovski said in response to concerns about AI hallucinations. “And I think the way to do that is to improve the data set.”

For example, Technovation projects need to have data on local issues to solve local problems.

“We work with orphanages across Vietnam, and the girls are very concerned that the standard image recognition models that are accessible do not recognize Vietnamese facial expressions,” said Chklovski. “And so they developed their own data set and trained it so that it could work for their population.”

SEE: Stay up-to-date on artificial intelligence with TechRepublic’s cheat sheet .

Women in AI: How businesses can encourage young women in tech

Despite years of progress in various sectors, the tech industry continues to grapple with a longstanding issue: the persistent underrepresentation of women.

Recent research shows that women hold about 26% of tech-related jobs in the U.S., despite women comprising nearly half of the labor force. Women make up 35% of all employees in computer system design and related services in the U.S.

Improving gender equity in the workplace reduces the chance that products, including generative AI, will show bias. A 2018 study from Deloitte found that when organizational leaders foster inclusion, 70% of workers report an increase in “respect, value, and longing,” as well as “psychological safety; and inspiration.”

Conversely, having fewer women in tech limits the pool of people who can help businesses find solutions to real problems. For example, projects worked on in Technovation groups often align with the UN General Assembly’s global goals .

To encourage young women to pursue or continue tech careers, businesses should look outside traditional college pipelines for the skills they want to hire, Chklovski said. Many conversations about AI in education are “myopic,” she suggested, because the technology moves too quickly.

“Prompt engineering is something else. People talk about it, and I think that that’s aiming too low,” she said. “Because by the time you develop the curriculum, you train the teachers, you deploy this at large scale, the technology has already changed. It’s more important to teach young people, especially, to build future-proof skills where lifelong learning is the key.”

Recruiters should search for products young women like the graduates of Technovation have already built and the problems they have solved. For example, Chklovski said, a Technovation team created an Uber-like app in 2010, before the proliferation of rideshare companies. Another group created an app that encouraged digital “focus time,” long before it was a common mantra that people might struggle with attentiveness in a flood of digital information.

She suggested businesses bring a project-first, hands-on approach to workplace training, encouraging people to pick a problem and learn to solve it rather than watching “extremely boring” training courses.

Businesses should also consider internships for young women who hold computer science degrees or have hand-on experience from beyond the top 20 universities, Chklovski noted.

“We have 11,000 alumna who are above the age of 18 — looking for internships, early career opportunities — and have a robust portfolio to show what they have built, what kinds of problems they have solved,” she said. “It takes away some of the uncertainties from hiring.”

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What Kamala Harris has said so far on key issues in her campaign

As she ramps up her nascent presidential campaign, Vice President Kamala Harris is revealing how she will address the key issues facing the nation.

In speeches and rallies, she has voiced support for continuing many of President Joe Biden’s measures, such as lowering drug costs , forgiving student loan debt and eliminating so-called junk fees. But Harris has made it clear that she has her own views on some key matters, particularly Israel’s treatment of Gazans in its war with Hamas.

In a departure from her presidential run in 2020, the Harris campaign has confirmed that she’s moved away from many of her more progressive stances, such as her interest in a single-payer health insurance system and a ban on fracking.

Harris is also expected to put her own stamp and style on matters ranging from abortion to the economy to immigration, as she aims to walk a fine line of taking credit for the administration’s accomplishments while not being jointly blamed by voters for its shortcomings.

Her early presidential campaign speeches have offered insights into her priorities, though she’s mainly voiced general talking points and has yet to release more nuanced plans. Like Biden, she intends to contrast her vision for America with that of former President Donald Trump. ( See Trump’s campaign promises here .)

“In this moment, I believe we face a choice between two different visions for our nation: one focused on the future, the other focused on the past,” she told members of the historically Black sorority Zeta Phi Beta at an event in Indianapolis in late July. “And with your support, I am fighting for our nation’s future.”

Here’s what we know about Harris’ views:

Harris took on the lead role of championing abortion rights for the administration after Roe v. Wade was overturned in June 2022. This past January, she started a “ reproductive freedoms tour ” to multiple states, including a stop in Minnesota thought to be the first by a sitting US president or vice president at an abortion clinic .

On abortion access, Harris embraced more progressive policies than Biden in the 2020 campaign, as a candidate criticizing his previous support for the Hyde Amendment , a measure that blocks federal funds from being used for most abortions.

Policy experts suggested that although Harris’ current policies on abortion and reproductive rights may not differ significantly from Biden’s, as a result of her national tour and her own focus on maternal health , she may be a stronger messenger.

High prices are a top concern for many Americans who are struggling to afford the cost of living after a spell of steep inflation. Many voters give Biden poor marks for his handling of the economy, and Harris may also face their wrath.

In her early campaign speeches, Harris has echoed many of the same themes as Biden, saying she wants to give Americans more opportunities to get ahead. She’s particularly concerned about making care – health care, child care, elder care and family leave – more affordable and available.

Harris promised at a late July rally to continue the Biden administration’s drive to eliminate so-called “junk fees” and to fully disclose all charges, such as for events, lodging and car rentals. In early August, the administration proposed a rule that would ban airlines from charging parents extra fees to have their kids sit next to them.

On day one, I will take on price gouging and bring down costs. We will ban more of those hidden fees and surprise late charges that banks and other companies use to pad their profits.”

Since becoming vice president, Harris has taken more moderate positions, but a look at her 2020 campaign promises reveals a more progressive bent than Biden.

As a senator and 2020 presidential candidate, Harris proposed providing middle-class and working families with a refundable tax credit of up to $6,000 a year (per couple) to help keep up with living expenses. Titled the LIFT the Middle Class Act, or Livable Incomes for Families Today, the measure would have cost at the time an estimated $3 trillion over 10 years.

Unlike a typical tax credit, the bill would allow taxpayers to receive the benefit – up to $500 – on a monthly basis so families don’t have to turn to payday loans with very high interest rates.

As a presidential candidate, Harris also advocated for raising the corporate income tax rate to 35%, where it was before the 2017 Tax Cuts and Jobs Act that Trump and congressional Republicans pushed through Congress reduced the rate to 21%. That’s higher than the 28% Biden has proposed.

Affordable housing was also on Harris’ radar. As a senator, she introduced the Rent Relief Act, which would establish a refundable tax credit for renters who annually spend more than 30% of their gross income on rent and utilities. The amount of the credit would range from 25% to 100% of the excess rent, depending on the renter’s income.

Harris called housing a human right and said in a 2019 news release on the bill that every American deserves to have basic security and dignity in their own home.

Consumer debt

Hefty debt loads, which weigh on people’s finances and hurt their ability to buy homes, get car loans or start small businesses, are also an area of interest to Harris.

As vice president, she has promoted the Biden administration’s initiatives on student debt, which have so far forgiven more than $168 billion for nearly 4.8 million borrowers . In mid-July, Harris said in a post on X that “nearly 950,000 public servants have benefitted” from student debt forgiveness, compared with only 7,000 when Biden was inaugurated.

A potential Harris administration could keep that momentum going – though some of Biden’s efforts have gotten tangled up in litigation, such as a program aimed at cutting monthly student loan payments for roughly 3 million borrowers enrolled in a repayment plan the administration implemented last year.

The vice president has also been a leader in the White House efforts to ban medical debt from credit reports, noting that those with medical debt are no less likely to repay a loan than those who don’t have unpaid medical bills.

In a late July statement praising North Carolina’s move to relieve the medical debt of about 2 million residents, Harris said that she is “committed to continuing to relieve the burden of medical debt and creating a future where every person has the opportunity to build wealth and thrive.”

Health care

Harris, who has had shifting stances on health care in the past, confirmed in late July through her campaign that she no longer supports a single-payer health care system .

During her 2020 campaign, Harris advocated for shifting the US to a government-backed health insurance system but stopped short of wanting to completely eliminate private insurance.

The measure called for transitioning to a Medicare-for-All-type system over 10 years but continuing to allow private insurance companies to offer Medicare plans.

The proposal would not have raised taxes on the middle class to pay for the coverage expansion. Instead, it would raise the needed funds by taxing Wall Street trades and transactions and changing the taxation of offshore corporate income.

When it comes to reducing drug costs, Harris previously proposed allowing the federal government to set “a fair price” for any drug sold at a cheaper price in any economically comparable country, including Canada, the United Kingdom, France, Japan or Australia. If manufacturers were found to be price gouging, the government could import their drugs from abroad or, in egregious cases, use its existing but never-used “march-in” authority to license a drug company’s patent to a rival that would produce the medication at a lower cost.

Harris has been a champion on climate and environmental justice for decades. As California’s attorney general, Harris sued big oil companies like BP and ConocoPhillips, and investigated Exxon Mobil for its role in climate change disinformation. While in the Senate, she sponsored the Green New Deal resolution.

During her 2020 campaign, she enthusiastically supported a ban on fracking — but a Harris campaign official said in late July that she no longer supports such a ban.

Fracking is the process of using liquid to free natural gas from rock formations – and the primary mode for extracting gas for energy in battleground Pennsylvania. During a September 2019 climate crisis town hall hosted by CNN, she said she would start “with what we can do on Day 1 around public lands.” She walked that back later when she became Biden’s running mate.

Biden has been the most pro-climate president in history, and climate advocates find Harris to be an exciting candidate in her own right. Democrats and climate activists are planning to campaign on the stark contrasts between Harris and Trump , who vowed to push America decisively back to fossil fuels, promising to unwind Biden’s climate and clean energy legacy and pull America out of its global climate commitments.

If elected, one of the biggest climate goals Harris would have to craft early in her administration is how much the US would reduce its climate pollution by 2035 – a requirement of the Paris climate agreement .

Immigration

Harris has quickly started trying to counter Trump’s attacks on her immigration record.

Her campaign released a video in late July citing Harris’ support for increasing the number of Border Patrol agents and Trump’s successful push to scuttle a bipartisan immigration deal that included some of the toughest border security measures in recent memory.

The vice president has changed her position on border control since her 2020 campaign, when she suggested that Democrats needed to “critically examine” the role of Immigration and Customs Enforcement, or ICE, after being asked whether she sided with those in the party arguing to abolish the department.

In June of this year, the White House announced a crackdown on asylum claims meant to continue reducing crossings at the US-Mexico border – a policy that Harris’ campaign manager, Julie Chavez Rodriguez, indicated in late July to CBS News would continue under a Harris administration.

Trump’s attacks stem from Biden having tasked Harris with overseeing diplomatic efforts in Central America in March 2021. While Harris focused on long-term fixes, the Department of Homeland Security remained responsible for overseeing border security.

She has only occasionally talked about her efforts as the situation along the US-Mexico border became a political vulnerability for Biden. But she put her own stamp on the administration’s efforts, engaging the private sector.

Harris pulled together the Partnership for Central America, which has acted as a liaison between companies and the US government. Her team and the partnership are closely coordinating on initiatives that have led to job creation in the region. Harris has also engaged directly with foreign leaders in the region.

Experts credit Harris’ ability to secure private-sector investments as her most visible action in the region to date but have cautioned about the long-term durability of those investments.

Israel-Hamas

The Israel-Hamas war is the most fraught foreign policy issue facing the country and has spurred a multitude of protests around the US since it began in October.

After meeting with Israeli Prime Minister Benjamin Netanyahu in late July, Harris gave a forceful and notable speech about the situation in Gaza.

We cannot look away in the face of these tragedies. We cannot allow ourselves to become numb to the suffering. And I will not be silent.”

Harris echoed Biden’s repeated comments about the “ironclad support” and “unwavering commitment” to Israel. The country has a right to defend itself, she said, while noting, “how it does so, matters.”

However, the empathy she expressed regarding the Palestinian plight and suffering was far more forceful than what Biden has said on the matter in recent months. Harris mentioned twice the “serious concern” she expressed to Netanyahu about the civilian deaths in Gaza, the humanitarian situation and destruction she called “catastrophic” and “devastating.”

She went on to describe “the images of dead children and desperate hungry people fleeing for safety, sometimes displaced for the second, third or fourth time.”

Harris emphasized the need to get the Israeli hostages back from Hamas captivity, naming the eight Israeli-American hostages – three of whom have been killed.

But when describing the ceasefire deal in the works, she didn’t highlight the hostage for prisoner exchange or aid to be let into Gaza. Instead, she singled out the fact that the deal stipulates the withdrawal by the Israeli military from populated areas in the first phase before withdrawing “entirely” from Gaza before “a permanent end to the hostilities.”

Harris didn’t preside over Netanyahu’s speech to Congress in late July, instead choosing to stick with a prescheduled trip to a sorority event in Indiana.

Harris is committed to supporting Ukraine in its fight against Russian aggression, having met with Ukrainian President Volodymyr Zelensky at least six times and announcing last month $1.5 billion for energy assistance, humanitarian needs and other aid for the war-torn country.

At the Munich Security Conference earlier this year, Harris said: “I will make clear President Joe Biden and I stand with Ukraine. In partnership with supportive, bipartisan majorities in both houses of the United States Congress, we will work to secure critical weapons and resources that Ukraine so badly needs. And let me be clear: The failure to do so would be a gift to Vladimir Putin.”

More broadly, NATO is central to our approach to global security. For President Biden and me, our sacred commitment to NATO remains ironclad. And I do believe, as I have said before, NATO is the greatest military alliance the world has ever known.”

Police funding

The Harris campaign has also walked back the “defund the police” sentiment that Harris voiced in 2020. What she meant is she supports being “tough and smart on crime,” Mitch Landrieu, national co-chair for the Harris campaign and former mayor of New Orleans, told CNN’s Pamela Brown in late July.

In the midst of nationwide 2020 protests sparked by George Floyd’s murder by a Minneapolis police officer, Harris voiced support for the “defund the police” movement, which argues for redirecting funds from law enforcement to social services. Throughout that summer, Harris supported the movement and called for demilitarizing police departments.

Democrats largely backed away from calls to defund the police after Republicans attempted to tie the movement to increases in crime during the 2022 midterm elections.

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• Open access
• Published: 01 August 2024

Population diversity control based differential evolution algorithm using fuzzy system for noisy multi-objective optimization problems

• Brindha Subburaj 1 ,
• J. Uma Maheswari 1 ,
• S. P. Syed Ibrahim 1 &
• Muthu Subash Kavitha 2  

Scientific Reports volume  14 , Article number:  17863 ( 2024 ) Cite this article

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The objective measurements of the real-world optimization problems are mostly subject to noise which occurs due to several reasons like human measurement or environmental factors. The performance of the optimization algorithm gets affected if the effect of noise is higher than the negligible limit. The previous noise handling optimization algorithms use a large population size or multiple sampling at same region which increases the total count of function evaluations, and few methods work for a particular problem type. To address the above challenges, a Differential Evolution based Noise handling Optimization algorithm (NDE) to solve and optimize noisy bi-objective optimization problems is proposed. NDE is a Differential Evolution (DE) based optimization algorithm where the strategies for trial vector generation and the control parameters of DE algorithm are self-adapted using fuzzy inference system to improve the population diversity along the evolution process. In NDE, explicit averaging based method for denoising is used when the noise level is higher than negligible limit. Extending noise handling method enhances the performance of the optimization algorithm in solving real world optimization problems. To improve the convergence characteristics of the proposed algorithm, a restricted local search procedure is proposed. The performance of NDE algorithm is experimented using DTLZ and WFG problems, which are benchmark bi-objective optimization problems. The obtained results are compared with other SOTA algorithm using modified Inverted Generational Distance and Hypervolume performance metrics, from which it is confirmed that the proposed NDE algorithm is better in solving noisy bi-objective problems when compared to the other methods. To further strengthen the claim, statistical tests are conducted using the Wilcoxon and Friedman rank tests, and the proposed NDE algorithm shows significance over the other algorithms rejecting the null hypothesis.

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Introduction.

Many real-world problems are formulated as optimization problems. While modeling the optimization problems, noise may arise from different roots and such problems are called noisy optimization problems. There are various sources from which noise may arise such as, measurement errors, incompleteness in data and environmental factors, due to which it is difficult to attain appropriate objective function value 1 , 2 , 3 . The impact of such noise factors in optimization problems can be, different objective value results over multiple evaluations for a same individual. The other impact of noise is, when the noise strength is high, population will contain large number of poor solutions, due to which the environment selection process is affected and deviates the search direction away from the true front 4 . Thus, when the effect of noise is high it may influence more in search process 5 . It is vital to consider noise factor while modelling the optimization problem. Including the noise factor along with objective function value while evaluation is presented in Eq.  1 .

where \(\varepsilon\) is the noise factor, \(\varepsilon \sim N(0,{\sigma }^{2},I)\) . The standard deviation component \({\sigma }^{2}\) denotes the noise strength and \(I\) is the identity matrix.

Evolutionary algorithms (EAs) mimic the natural selection and genetic inheritance principles. EA samples a population of candidate solutions, and new solutions are generated through selection, mutation and crossover operations. Promising solutions are selected for the next generation. EAs are robust in handling noisy optimization problems. The performance of EA may deteriorate when level of noise is high. Few notable draw backs of EA include fine tuning the control parameters associated with the algorithm according to the problem and premature convergence.

Real-world Engineering optimization problems mostly requires optimization of multiple objectives simultaneously. These problems are classified as multi-objective optimization problems (MOPs). Mathematical representation of a MOP is given by Eq.  2 ,

where, \(Y=[{Y}_{1}, {Y}_{2},.., {Y}_{d}]\) is a vector of \(d\) decision variables subject to boundary constraints and \(F(Y)\) is the objective vector with \(m\) objective functions. The objective of multi-objective optimization problem is to attain a set of pareto optimal solutions that exhibits good convergence and diversity characteristics.

Evolutionary algorithms are one among the widely used methods to solve such multi-objective optimization problems for the past two decades, since it is simple and much prior knowledge is not required and has a greater potential to perform global search. Nondominated sorting genetic algorithm II (NSGA-II) 6 , Strength pareto evolutionary algorithm (SPEA2) 7 are popular elitism based multi-objective evolutionary algorithms. As stated earlier, the performance of EA can be affected if the noise level is high.

The existing techniques designed for dealing with such optimization problems with noise factor can be categorized in to averaging, ranking and modeling based methods. Where, the averaging based methods can be categorized as explicit and implicit averaging methods. Explicit averaging method evaluates a solution for multiple times, and the average value is considered as the objective function value. In the implicit averaging model, size of the population is increased for reducing the impact of noise. Ranking based optimization methods are grouped in to probability ranking and clustering based ranking. In the probabilistic ranking method, the process of selecting fitter individuals is modified. Instead of the conventional dominance relation based selection operator, the probabilistic dominance factor is used to mitigate the effect of noise which estimates the dominance probability amongst two solutions. The clustering ranking method of noise handling is based on estimation of clustering radius in order to select fitter solutions. In the modeling based method of handling noisy optimization problems, a model is derived based on a solution set where the impact of noise could be less while compared to its effect on a single solution.

Given such wide range of algorithms to handle noisy optimization problems, the limitations associated with each of the methods are also to be considered. In general, averaging based methods are computationally expensive, since the function evaluations count it takes is high. In ranking based optimization methods, the results may be inaccurate since the dominance based ranking is replaced with probabilistic or other alternate techniques. Modeling method based algorithms may not be applicable to a wide range of optimization problems and can be suitable in solving problem with specific characteristics.

From the above study, we propose a Differential Evolution based noise handling optimization algorithm (NDE) to optimize noisy bi-objective optimization problems. The optimizer used is FAMDE-DC 8 as it is robust and efficient in handling optimization problems of varied characteristics. The main contributions include:

Differential evolution based noise handling optimization algorithm (NDE) is developed using the FAMDE-DC algorithm as the base optimizer combined with noise handling method.

Adaptive switching technique 9 is extended through which denoising method is applied only when the noise factor is high. Explicit averaging based method of denoising is used for handling noise.

To improve the exploitation properties of the NDE algorithm, a restricted local search technique is proposed.

Experimenting the performance of NDE algorithm. Benchmark optimization problems from DTLZ and WFG suite are used with noise inclusion. Modified inverted generational distance and the hypervolume indicators are utilized for performance evaluation. Results are compared with other algorithms and further non-parametric statistical test is conducted to strengthen the findings.

The paper is organized as follows, the related work is discussed in section “Related works”, the FAMDE-DC optimizer and the technique to measure the noise strength is given under section “Preliminaries”. In section “proposed algorithm”, the proposed NDE algorithm is detailed. In section “experimentation”, the experimental setup, test problems, performance metrics, results are presented. The conclusions are given in section “conclusions”. The notations used in the research and its expansion are listed in Table 1 .

Related works

Differential Evolution algorithm 10 is simple to extend, robust single objective optimization algorithms. Several algorithms by extending DE to solve MOPs are presented by various researchers. In 11 , authors introduced DE algorithm based on pareto dominance approach suitable to optimize multi-objective problems and the algorithm is named as Paret-frontier differential evolution algorithm (PDE). In Differential evolution for multi-objective optimization (DEMO) 12 , the base DE algorithm is combined with pareto ranking and crowding distance based sorting methods suitable to be applicable to optimize multi-objective optimization problems. The DEMO algorithm is further extended by 13 to solve many objective optimization problems by using correlation-based ordering of the objectives for the selection of conflicting objectives subset and they named it as \(\propto\) -DEMO. In 14 , the authors have presented a multi-objective self-adaptive differential evolution (MOSADE) in which the non-dominated solutions are retained using external elitist archive and further diversity is improved using crowding entropy measure. Differential evolution based multi-objective optimization by controlling the population diversity through self- adaptation of strategies used for trial vector generation and the control parameters using fuzzy system is achieved by FAMDE-DC algorithm 8 . In 15 , authors have proposed a self-adaptive Trajectory optimization method used to optimize the problem associated in the UAV based mobile edge computing system. Differential Evolution algorithm is improved using a distance indicator and two stage mutation strategy suitable to optimize multimodal multi-objective problems 16 . The above works show the importance and potential of the algorithms in solving optimization problems.

Some of the existing multi-objective evolutionary algorithms developed to handle noisy optimization problems are summarized. In 16 , authors have used iterative resampling procedure to lower the effect of noise. They have adapted a varying number of samples for a solution depending on the current noise factor in the search space. Two approaches are used for uncertainty reduction, such as resampling and increasing population size 17 which is applied to solve optimization problem in feedback control of combustion. Explicit averaging and modeling based method of denoising 9 is used to reduce the noise and the authors have proposed adaptive switch strategy to choose the noise treatment and type based on effect of noise. The population size is increased or decreased based on the observed objective function values in the predefined number of iterations 18 . Population size control based evolutionary strategy named pcCMSA-ES 19 , where linear regression and hypothesis test are used to detect noise effect, upon which the population size is varied. The learning rate parameter is adapted based up on the noise ratio 20 , the empirical results of this learning rate adaptation when compared to resampling or increasing the size of samples is better but, the convergence rate is not optimal.

To reduce noise, the authors have used stochastic and significance based dominance methods for solution selection 21 . Probability based ranking is used for selecting the fitter solutions to handle noise 22 . Clustering based ranking scheme 23 is used to handle noisy optimization problems. Algorithm based on restricted Boltzmann machine is used to build the probabilistic model and is hybridized with particle swarm algorithm to handle noisy optimization problems 24 . Regularity model is combined with NSGA-II algorithm for denoising 25 . Adaptive switch strategy 9 is introduced through switch and select amongst the denoising techniques such as averaging and modeling methods based on the noise strength measurements. In 26 , authors have used radial basis function networks as denoising method.

In Filters based NSGA-II (FNSGA-II) 27 , mean and wiener filters are included with optimization algorithms to handle noise in images and signals. These filters help to reduce the noise factor balancing convergence and diversity. A noise handling method for surrogate assisted evolutionary algorithms 28 by using radial basis function is proposed. Further, sampling strategies are chosen based on the convergence and diversity characteristics using adaptive switch technique. A Gaussian and Regularity model based NSGA-II algorithm named GMRM-NSGA-II is suggested to handle noisy multi-objective optimization problems. Population is divided to subpopulations and the above two models are applied to each of the population to improve convergence and diversity 30 .

The summary of literature review is given in Table 2 .

Preliminaries

The base optimizer FAMDE-DC and the technique to measure the noise strength is presented in this section.

FAMDE-DC algorithm

FAMDE-DC 8 is a multi-objective optimization algorithm. It is a Differential Evolution (DE) based algorithm. Crossover rate \((CR)\) value, which is a vital control parameter is self-adapted by using fuzzy system in order to improve the diversity among population. A pool of strategies for trial vector generation 29 is used to generate trial vectors and the strategies are self-adapted based on their success index in previous generations, which indicates how successful a strategy is in generating solutions which are entering in to successive generation after selection process. The selection of solutions that are fitter is performed through fast non-dominated sorting 6 , improved with controlled elitism 30 and dynamic crowding distance techniques 31 . Algorithm 1 shows the steps in FAMDE-DC algorithm and is detailed below.

Initialization

The initial population set of size ( \(NP\) ) is generated randomly covering the search space boundary limits. Next the control parameters \(CR\) and \(F\) of the DE algorithm are initialized. The learning period ( \(LE)\) is set as 50.

Fitness evaluation

Objective function value is estimated for all the solutions in the population set.

Crossover rate adaptation

The control parameter crossover rate is adapted using fuzzy inference system to control the diversity of population. After the fitness function evaluation, the current generation population diversity \((popdiversity)\) estimated using distance to average point technique 32 . Reference diversity variable \(\left(refdiversity\right)\) is set at 0.15. The difference between the current population diversity and reference diversity is estimated and given as input parameter to the single input/output fuzzy inference system. The fuzzy system maps the input with the implication rule set generated and it outputs the changes required to be made in the crossover rate in order to improve the population diversity. This change factor is used in crossover rate value generation, thus the crossover rate is self-adapted.

Trial vector generation

There are various strategies widely used in DE based algorithms for trial vector generation. The performance of the strategies varies among the optimization problems to which it is applied. Their performance characteristics even differs during the various stages of evolution, thus choosing a strategy for solving a particular optimization problem based on trial and error method is time consuming. Thus, a strategy pool with four trial vector generation strategies is extended 29 . During the initial learning period \((LE)\) the strategies are chosen with equal probability and later the strategies are chosen based on its success index value. The success index is an estimation of the percentage of solutions generated using a particular strategy, successfully chosen for the next generation after the selection process. The control parameter value crossover rate \((CR)\) is generated through normal distribution \(N(Mean,Std)\) , where the crossover rate mean \((CRm)\) is initially set at 0.5 and later is adapted using the population diversity and success index. The value of standard deviation set at 0.1. The scaling factor \((F)\) is generated using the normal distribution with mean and standard deviation values 0.5 and 0.3 respectively.

The trial and the target vectors are combined in to population size of \(2NP\) solutions. Fast non-dominated sorting 6 technique is used to split the solutions in to front levels based on non-domination factor. Solutions are selected based on controlled elitism 30 and dynamic crowding distance 31 techniques which ensures solutions from all the fronts are selected in order to improve the lateral diversity of the pareto front.

Measuring noise strength

Resampling is a common method used to measure and quantify the strength of the noise. The steps involved in estimation of noise strength is given below 9 , 33 . The very first step in measuring noise strength involves, estimating the count of resampling solutions which are utilized to observe noise strength which is estimated using the formula in Eq.  3 .

where, the variable \(NP\) is the size of the population set, the resampling ratio is given by \({\lambda }_{res}\) and this parameter limits the percentage of solutions that are to be re-evaluated. The count of solutions to be resampled is given by \({N}_{res}\) and \({f}_{pro}\) is the probabilistic based rounding function.

After estimating the count of resampling solutions, the second step is to choose randomly the \({N}_{res}\) number of solutions to be resampled. The third step is resampling the solutions and the technique of resampling differs based on the usage of explicit and implicit averaging denoising method. Upon resampling we will be left with two sets. For instance, let the solution set with noise inclusion is \({x}_{i}{\prime}={x}_{i}+N(0,{\sigma }^{2})\) , where \(i=\{\text{1,2},..d\}\) and \(d\) represents the problem dimension. The sets before and after resampling are: \({\mathcal{L}}_{1}=\{{\widetilde{f}}_{1}, {\widetilde{f}}_{2},\dots ,{\widetilde{f}}_{{N}_{re}},{\widetilde{f}}_{{N}_{re}+1},\dots {\widetilde{f}}_{N}\}\) and \({\mathcal{L}}_{2}=\{{\widetilde{f}}_{1}{\prime}, {\widetilde{f}}_{2}{\prime},\dots ,{\widetilde{f}}_{{N}_{re}}{\prime},{\widetilde{f}}_{{N}_{re}+1},\dots {\widetilde{f}}_{N}\}\) .

The fourth step is calculation of rank change, for each of the solutions that are resampled and is given in Eq.  4 .

where, the variable \(rank\left({\widetilde{f}}_{i}\right)\) and \(rank\left({\widetilde{f}}_{i}{\prime}\right)\) is the rank of solution in the set \(\mathcal{L}\) . The \(\text{sign}()\) function returns + 1/-1/0 based on the whether argument is positive/negative/otherwise, this function aids in eliminating the change of rank created by the resampling solution. The rank change fall within the range: \(\left|{\Delta }_{i}\right| \in \{\text{0,1},2,..2N-2\}\) .

The last step is measuring the noise strength \(s\) and is given in Eq.  5 .

where, in the above equation rank change \({\Delta }_{i}\) is compared with variable \({\Delta }_{\theta }^{lim}\) which is limit based on the acceptance threshold \(\theta\) , and it indicates \(\theta /2th\) percentile in the sequence of possible change of rank \({\mathbb{S}}=\{\left|1-R\right|,\left|2-R\right|, \dots ,|2N-1-R|\}\) . The steps involved include generation of the \({\mathbb{S}}\) sequence, next the sequence \({\mathbb{S}}\mathbb{^{\prime}}\) is obtained by arranging \({\mathbb{S}}\) and at last \(\theta /2th\) percentile is calculated from sequence \({\mathbb{S}}\mathbb{^{\prime}}\) . \(1\) is the indicator function and if its argument is matched correct then + 1 is returned else 0 is returned.

Thus, the strength of the noise given by variable \(s\) gives the range of noise in the objective function. If the result \(s<0\) then noise is within the limit of acceptance else, the resampled solutions have contributed to the change in rank beyond the acceptable limits and noise has huge effect over objective function.

Proposed algorithm

The proposed Differential evolution based noise handling optimization algorithm (NDE) is detailed in this section. The algorithm and flowchart are given below in Algorithm 2 and Fig.  1 respectively.

Flowchart representingthe proposedNDE algorithm.

Initial population set of size \(NP\) is generated randomly.

Trial vector is generated using a pool of strategies for trial vector generation 8 , 29 as used in the base optimizer FAMDE-DC. The strategies are adapted based on success index and crossover rate is adapted using fuzzy system.

Noise estimation and reduction

The effect or strength of noise is estimated through techniques as discussed in section “ Measuring noise strength ” with following modifications suitable to be applicable multi-objective optimization problems. The trial vector of \(NP\) solutions are evaluated with fast non-dominated sorting technique 6 and categorized in to non-dominated sets or fronts based on the domination count. The fitness of each solution is thus estimated using the fast non-dominated sorting based technique. The solutions used to measure the strength of noise for resampling is selected randomly from first front and, if sufficient solutions are not available in first front, then solutions from subsequent fronts are chosen till the required \({N}_{re}\) solutions are selected. If the estimated noise strength given by variable \(s\) is less than 0, then the effect of noise is minimal and it is not required to denoise. Else, the noise is to be reduced using denoising method. The process for noise strength measurement and reduction is given in Algorithm 3.

Noise strength measurement and reduction.

Explicit averaging-based method for denoising is applied to reduce strength of noise. This method uses resampling technique to denoise the values of objective functions for the selected solutions. In general, the resampling technique is applied for all the solutions for a specified count and averages the attained values as value for objective function. Selecting a solution set and resampling solutions in this set has shown significant improvement in the algorithm’s performance in denoising, and function evaluations utilized for resampling is thus wisely used through this method 9 .

In the proposed NDE algorithm, set of solutions are selected for resampling and each solution in the set are resampled twice. Fast non-dominated sorting is applied to the trial vector and solutions are categorized as fronts. The solutions listed in first front are set of non-dominated solutions. The solutions for resampling are selected from the first front. During the early stage of evolution, the number of solutions in the first front will be less and at later stage of evolution, the non-dominated solutions in the first front will be more. Thus, the solutions for resampling are chosen from the non-dominated solution set by forming a hyper box 9 using the ideal point \(({pop}_{min})\) and nadir point \(({pop}_{max})\) . These ideal and nadir points are chosen as in Eq.  6 .

\(\forall i\in (\text{1,2},\dots ,m)\) , where \(m\) indicates number of objectives, \(R\) is the count of non-dominated solutions in first front retrieved after performing non-dominated sorting. The solutions inside the hyper box are chosen for resampling.

To select solutions for next generation, target and the trial vectors are combined to form a set of \(2NP\) solutions, where \(NP\) is the size of the population. This combined population set is subject to fast non-dominated sorting. The required \(NP\) solutions are selected from first front, which is the set of non-dominated solutions. If sufficient number of solutions are not available in first front, then solutions from subsequent fronts are used. If more than required number of solutions are available in a front then, Dynamic Crowding Distance (DCD) 30 technique is used to select the required solutions from the front. DCD ensures maintaining uniform diversity across the selected solutions.

• Local search

The function evaluations consumed during the explicit averaging-based method of denoising may reduce the convergence properties of the algorithm. Moreover, the crossover rate \(CR\) self-adaptation using fuzzy system through population diversity control may also affect the convergence. Thus, to improve the convergence, exploitation of solutions in promising search region is done using a restricted local search algorithm. Local search based optimization method searches for a locally better solution compared to the current solution chosen for exploitation in its proximity, and if one such solution is found it is added to the population which improves the convergence rate.

In the present NDE algorithm, we propose a restricted local search technique. The local search is performed in regular prespecified intervals \({LS}_{Int}\) and for a limited number of function evaluations \({LS}_{Feval}\) , through which exploitation at larger rate is restricted. The current population best solution is chosen from the first front (the set of non-dominated solutions). Search is performed in the proximity of chosen solution for \({LS}_{Feval}\) times and if a solution better (dominates) than the current population best solution or better than any solution in first front is identified through the search, then the identified solution replaces a random solution in the population. Thus, this restricted local search procedure, applied once in every \({LS}_{Int}\) generation tries to find a better solution through exploitation and improve the convergence rate. The \({LS}_{Int}\) and \({LS}_{Feval}\) values are selected based on trial and error basis and best parameter values are selected for further experimentation. The restricted local search method is given in Algorithm 4.

Restricted local search.

Experimentation

To evaluate the performance of the proposed NDE algorithm, DTLZ 34 and WFG 35 test problems are taken. The above test problem set are multi-objective unconstrained optimization problems and varying noise levels (0.1, 0.2, 0.5) is used for experimentation, and is detailed in section " Test problems ". The performance metrics used for experimentation are modified inverted generational distance (IGD + ) 36 and hypervolume (HV) 37 , which helps to assess both convergence and diversity properties of the attained solution set and the metrics are detailed in section " Performance metrics ".

The performance of proposed NDE algorithm is compared with five multi-objective evolutionary algorithms namely, two stage evolutionary algorithm (TSEA) 9 , two archive algorithm (Two_Arch2) 38 , regularity model in NSGA-II (RM-NSGA-II) 25 , Noise-tolerant Strength Pareto Evolutionary Algorithm (NTSPEA) 2 , rolling tide evolutionary algorithm (RTEA) 39 and Filters based NSGA-II (FNSGA-II) 27 . Except Two_Arch2 all other algorithms are developed to solve noisy optimization problems. The initialization of values for the various parameters used in proposed NDE algorithm are listed in Table 3 .

Test problems

Sixteen test problems are used to evaluate the performance of the proposed algorithm. All the problems are multi-objective unconstrained optimization problems. From DTLZ test suite 34 seven problems (DTLZ1 to DTLZ7) are taken and from WFG test suite 35 nine problems (WFG1 to WFG9) are taken. The number of decision variables and objectives can be scaled up to required numbers. These sixteen problems exhibits different properties (concave, multimodal, biased, etc.) make it suitable to be used to test the performance of the proposed optimization algorithm. Three objective problems are considered for investigation.

The properties of these problems are listed in Table 4 . The significance of using above problems in evaluating optimization algorithms is the availability of true pareto optimal front. Further, in DTLZ problems the number of objectives is scalable, test problems DTLZ5 and DTLZ6 have degenerated pareto fronts and distance functions are added in all the DTLZ test problems. WFG Problems include complex characteristics like many problems are non-separable, we may also observe few problems are deceptive, problems where pareto fronts are mixed shape. The availability of problems with such complex characteristics and availability true optimal fronts for the benchmark problems like above, significantly contributes in conducting the experimentation in the optimization field.

Performance metrics

The performance of the proposed algorithm is investigated using two performance metrics, modified Inverted Generational Distance (IGD + ) 36 and HyperVolume (HV) 37 .

IGD + performance metric is a modified inverted generational distance metric that quantifies both convergence and diversity characteristic of an optimization algorithm. It helps to estimate the distance between the attained and true pareto fronts, lesser the IGD + value indicates better the attained solutions and is calculated as given in Eq.  7 .

where, \(A\) represents the reference points in true front, \(B\) represents the solutions in the attained pareto front. \(d(x,B)\) represents the nearest distance from \(x\) to attained front solutions, and this distance calculation from \(x\) to a solution \(y\) in \(B\) is estimated as, \(d\left(x,y\right)=\sqrt{\sum_{j=1}^{n}\text{max}{({y}_{j}-{x}_{j},0)}^{2}}\) and \(n\) represents the number of objectives.

HV performance metric helps to evaluate the convergence and diversity properties of the obtained solutions. HV calculates the hypervolume between the attained front and a given reference point. Formula to calculate HV is given in Eq.  8 . A higher HV value indicates that better solutions are attained.

where variable \(S\) represents attained solution set and \(Le\) indicates the Lebesgue measure. Variable \(n\) denotes the number of objective functions. \(Re=({re}_{1}, {re}_{2},..,{re}_{m})\) is a vector of maximum reference point value on every objective function set at (1,1). Before calculating HV metric value, the objective functions are normalized using Min–max normalization.

Results and discussion on DTLZ problems

The mean IGD + value results of DTLZ problems are given in Table 5 and the pareto fronts are given in Fig.  2 . The results of HV Metric is listed in Table 6 . The overall performance of NDE algorithm results is better when compared to all algorithms. DTLZ1 test problem is a linear and multi-modal test problem and chances of getting stuck at local optimal solutions are higher for problems of such characteristics, the obtained pareto front is shown in Fig.  2 a. NDE algorithms performance is best followed by TSEA algorithm and the other algorithms. This is because of the fuzzy system based control parameter adaptation to regulate the population diversity and the applied denoising method. DTLZ2 is a concave, unimodal problem and the possibility of noise affecting its performance is high. NDE outperforms and the results are better compared to the other algorithms, and the respective attained front is shown in Fig.  2 b. The effectiveness of averaging based denoising method is evident through the results. The characteristic of DTLZ3 test problem is multimodal as well and the performance of the proposed NDE algorithm is better, and the attained front is given in Fig.  2 c.

Obtained pareto fronts for DTLZ problems. ( a ) Obtained pareto fronts for DTLZ1 problem. ( b ) Obtained pareto fronts for DTLZ2 problem. ( c ) Obtained pareto fronts for DTLZ3 problem. ( d ) Obtained pareto fronts for DTLZ4 problem. ( e ) Obtained pareto fronts for DTLZ5 problem. ( f ) Obtained pareto fronts for DTLZ6 problem. ( g ) Obtained pareto fronts for DTLZ7 problem.

DTLZ4 test problem is a modified one from DTLZ2 and the solutions are usually densely populated near to the planes. The performance of NTSPEA algorithm is better followed by NDE and the other methods. It can be observed that the NTSPEA algorithm works by associating survival time factor for all the solutions in the population and DTLZ4 problem is sensitive towards the initial population, and the respective attained front is given in Fig.  2 d. The characteristics of test problems DTLZ5, DTLZ6 and DTLZ7 are irregular and the pareto front for DTLZ7 is discontinuous, the obtained pareto fronts for these problems are illustrated in Fig.  2 e, f and g respectively. The proposed NDE algorithm attains better results for all the above problems. The IGD + and HV values evolution across the function evaluations for DTLZ2 and DTLZ7 test problems are given in Fig.  3 a and b respectively.

( a ) IGD + values across the function evaluations for DTLZ2 test problem. ( b ) HV values across the function evaluations for DTLZ7 test problem.

Results and discussion on WFG problems

WFG test problems are relatively complex when compared to the DTLZ problems. The mean values of the IGD + and HV performance metrics are given in Tables 7 , 8 respectively. The attained pareto fronts for the WFG problems are given in Fig.  4 . WFG1 test problem is a complex test problem with mixed and biased characteristics. The performance of the proposed NDE algorithm is better than other algorithms taken for comparison, and is at par with performance of NTSPEA algorithm, the obtained pareto front for the same is illustrated in Fig.  4 a. WFG2 is a multimodal test problem, and the performance of NDE is the best, and the respective front is given in Fig.  4 b. WFG3 is a linear, non-separable test problem and the attained solutions and the mean IGD + and HV values are better than other methods, and the pareto front is given in Fig.  4 c.

Obtained pareto fronts for WFG problems. ( a ) Obtained pareto fronts for WFG1 problem. ( b ) Obtained pareto fronts for WFG2 problem. ( c ) Obtained pareto fronts for WFG3 problem. ( d ) Obtained pareto fronts for WFG4 problem. ( e ) Obtained pareto fronts for WFG5 problem. ( f ) Obtained pareto fronts for WFG6 problem. ( g ) Obtained pareto fronts for WFG7 problem. ( h ) Obtained pareto fronts for WFG8 problem. ( i ) Obtained pareto fronts for WFG9 problem.

The characteristic of WFG4 problem is that it is multi-modal, with multiple local optimal solutions. The population diversity control and the appropriate denoising method aid in escaping from such locally optimal solutions and it is evident through the obtained results, and front as given in Fig.  4 d. WFG5 problem is of deceptive characteristic and is a separable problem and the attained concave pareto front and the mean IGD + and HV values are better, and the front for the problem is illustrated in Fig.  4 e. WFG6, WFG8 and WFG9 test problems are non-separable test problems and WFG9 is multi-modal as well. For the test problem WFG6, the performance of NDE and RTEA are at par, followed by the other algorithms, and the attained pareto front is illustrated in Fig.  4 f. For WFG8 and WFG9 test problems, the results of NDE algorithm are promising, and the obtained pareto fronts for these problems are given in Fig.  4 h and i respectively. The characteristic of WFG7 is it is unimodal and a separable one and the results of the proposed NDE algorithm is best, and the pareto front is given in Fig.  4 g. The IGD + and HV values evolution across the function evaluations for WFG9 and WFG2 test problems are given in Fig.  5 a and b respectively.

( a ) IGD + values across the function evaluations for WFG9 test problem. ( b ) HV values across the function evaluations for WFG2 test problem.

Population effect and population diversity analysis

Population plays a vital role in search and optimization algorithms. The initial population is randomly generated within the variable bounds. Along the evolution, fitter solutions are selected and used that helps to attain an optimal or pareto-optimal solutions. In the proposed research, the population diversity plays a key role. The population diversity is controlled adaptively in order to improve the search performance. The solutions in the population must be diverse enough during the initial stages of evolution to have a better exploration over the search space and over the evolution and at later stages the population diversity is to be low to have better exploitation. Population that aids in balancing the explore-exploit cycle is required to attain optimal or pareto optimal solutions. The effect of population diversity in the present research is presented below and the explore-exploit cycle analysis is presented in the section " Exploration–Exploitation analysis ".

To perform population diversity analysis, diversity of the population is calculated as given in Eq.  9 , “distance to average point” measure 32 . This measure includes population size, dimension and search region of variables, thus this measure used for population diversity estimation.

where, \(PP\) represents population with size \(NP\) . \(\left|LenDia\right|\) is the diagonal length of the search space, \({d}_{i}\) is the problem dimension. \({y}_{ij}\) is \({j}^{th}\) value of \({i}^{th}\) solution, \({\overline{y}}_{j}\) is the \({j}^{th}\) value of average point \(\overline{y }\) .

Figure  6 a and b shows the diversity graph of the population along the iterations for two problems, from which it is evident that, during the initial stages of evolution higher diversity is observed to improve the exploration and in later stage diversity value is lesser to improve exploitation.

( a ) Population diversity adaptation along the iterations for DTLZ1 test problem. ( b ) Population diversity adaptation along the iterations for WFG7 test problem.

Exploration–Exploitation analysis

To analyse the balancing property of exploration–exploitation cycle, the exploration and exploitation rate along the evolution is calculated using Eqs.  10 and 11 40 .

where, \(popdiversity\) indicates the population diversity and is calculated using Eq.  9 .

Exploration–exploitation graph for two problems is given in Fig.  7 a and b. It can be observed that, exploration starts with a higher value and during later stages it decreases and the exploitation begins with a smaller value and upon evolution its value is increased in final stages, which ensures the balance between the cycles of exploration and exploitation.

( a ) Exploration–exploitation graph for DTLZ5 test problem. ( b ) Exploration–exploitation graph for WFG2 test problem.

Statistical test results

Statistical analysis through Wilcoxon signed rank test and Friedman test 41 are performed to further validate the performance of NDE.

Wilcoxon signed rank test is a pairwise comparison test, used to find whether there is significant difference between two algorithms. The test is performed using IBM SPSS package and the significance level \(\alpha\) is set to 0.05. \(\rho\) -value is computed through the above test and if the obtained \(\rho\) -value is less than \(\alpha\) , the null hypothesis can be rejected and it implies there is significance difference among the algorithms taken for evaluation. The test results using the mean HV value results with \(\sigma\) value 0.5 obtained through various algorithms for DTLZ problems is presented in Table 9 . The \(\rho\) -value is less than 0.05 (significance level) and the null hypothesis can be rejected. It is evident that the proposed NDE algorithm is significantly better when compared to other algorithms taken for comparison.

Table 10 gives the Wilcoxon signed rank test results using the mean IGD + value results with.

\(\sigma\) value 0.5 obtained through various algorithms for WFG test problems. It is evident that the NDE algorithm is significantly better than the other algorithms taken for comparison as the \(\rho\) -value is less than 0.05.

The competence of the proposed NDE algorithm is also statistically analysed using Friedman rank test, which is a multiple comparison test. Joint analysis of algorithms taken for comparison is performed through the above test. The test is performed using IBM SPSS package and the significance level \(\alpha\) is set to 0.05. \(\rho\) -value is computed through the above test and if the obtained \(\rho\) -value is less than \(\alpha\) , it implies there is significance difference among the algorithms taken for evaluation.

Table 11 gives the Friedman rank test results using the mean IGD + value results with.

\(\sigma\) value 0.5 obtained through various algorithms for DTLZ problems.

The proposed NDE algorithm is ranked first in the above test and the \(\rho\) -value obtained is 0.0001 which is less than the significance level \(\alpha =0.05\) , which proves the significance of the proposed optimization algorithm.

Table 12 gives the Friedman rank test results using the mean HV value results with \(\sigma\) value 0.5 obtained through various algorithms for WFG problems.

The proposed NDE algorithm is again ranked first in the above statistical test and the \(\rho\) -value obtained is 0.00008 which is less than the significance level \(\alpha =0.05\) , shows the significance of the proposed optimization algorithm.

Experimentation on CEC 2017 test problems

To further investigate the effectiveness of the proposed Differential Evolution based Noise handling Optimization algorithm (NDE), experimentation is conducted on CEC 2017 test problems 42 which are single objective optimization problems. To perform this study, the selection process in the proposed NDE algorithm is changed suitable to handle single objective optimization problems. The trial vector generated after the crossover operation is subject to fitness function evaluation. The fitness function values of the solutions in target and trial vectors are compared and the better solution is chosen as candidate for the next generation.

CEC 2017 test suite comprises 29 test problems (Test problem F2 has been removed from the CEC 2017 suite). The parameter settings are followed as given in the technical report and are given in Table 13 .

The result analysis is performed by calculating error value, which is the difference between the best objective function value that is attained in a run and the true optimal value. This error value is calculated for all the 51 runs and the mean and standard deviation values are recorded. The results are compared with two other algorithms Effective Butterfly Optimizer using Covariance Matrix Adapted Retreat phase (EBOwithCMAR) 43 and jSO 44 which has secured first and second rank in the CEC 2017 competition and are presented in Table 14 . The best results are highlighted in boldface, and it can be observed that the proposed NDE algorithm performs better in 19 functions, similar performance in 3 functions and inferior performance in 7 functions. This shows the significance of the NDE algorithm, with its capability of population diversity control through strategy adaptations, the restricted local search and noise handling techniques exhibits a robust performance.

Limitations

The main limitations of the research include, experimenting on a number of benchmark problems with varied and complex characteristics and not conducting experimentation on a real-world optimization problem 45 , 46 , 47 . The widely considered limitation in applying an optimization algorithm to a real-world problem will be parameter fine tuning according to the problem. But in the proposed NDE algorithm the major parameters like crossover rate and trial vector generation strategies are self-adapted according to the population domain of the problem. Thus, finetuning the other associated parameters may be a challenge. Through the above test results the robustness of the proposed NDE algorithm in solving a wider range of problems is evident. Further improvements will be studied in future work, by integrating multiple denoising models, fine tuning algorithm to handle constrained optimization problems and conducting experiments on real-world optimization problem.

Conclusions

In the present work, Differential Evolution based Noise handling Optimization algorithm (NDE) to optimize noisy bi-objective optimization problems is proposed. FAMDE-DC is used as the base optimizer, where fuzzy system is extended to adapt crossover rate in order to control the population diversity. Adaptive switching technique is extended to test whether the noise ratio is within the acceptable limits or not. If the noise ratio is exceeding the prespecified limits, then explicit averaging based denoising method is applied. To further improve the convergence characteristics, a restricted local search procedure is applied in prespecified intervals. To evaluate the performance of proposed NDE algorithm, DTLZ and WFG test problems are used by including noise of different levels. The modified Inverted Generational Distance (IGD + ) and Hypervolume (HV) are chosen as the performance indicators. The attained results for most of the above problems using NDE algorithm are better than SOTA algorithms that are taken for comparison and shows the effectiveness of NDE algorithm in handling noisy optimization problems. The results related to test problem DTLZ4 are not best, the pareto optimal solutions for the problem are densely populated near to the planes and the convergence characteristics of the attained solutions are to be improved, which shows the future direction of improving the NDE algorithm. Further, non-parametric statistical tests namely Wilcoxon signed rank test and Friedman rank tests are conducted on the attained results, and the test results shows the significance of the suggested NDE algorithm over the other algorithms taken for comparison.

Data availability

The datasets used and analysed during the current study available from the corresponding author on reasonable request.

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Subburaj, B., Maheswari, J.U., Ibrahim, S.P.S. et al. Population diversity control based differential evolution algorithm using fuzzy system for noisy multi-objective optimization problems. Sci Rep 14 , 17863 (2024). https://doi.org/10.1038/s41598-024-68436-1

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Real World Problem-Solving

Real world problem-solving (RWPS) is what we do every day. It requires flexibility, resilience, resourcefulness, and a certain degree of creativity. A crucial feature of RWPS is that it involves continuous interaction with the environment during the problem-solving process. In this process, the environment can be seen as not only a source of inspiration for new ideas but also as a tool to facilitate creative thinking. The cognitive neuroscience literature in creativity and problem-solving is extensive, but it has largely focused on neural networks that are active when subjects are not focused on the outside world, i.e., not using their environment. In this paper, I attempt to combine the relevant literature on creativity and problem-solving with the scattered and nascent work in perceptually-driven learning from the environment. I present my synthesis as a potential new theory for real world problem-solving and map out its hypothesized neural basis. I outline some testable predictions made by the model and provide some considerations and ideas for experimental paradigms that could be used to evaluate the model more thoroughly.

1. Introduction

In the Apollo 13 space mission, astronauts together with ground control had to overcome several challenges to bring the team safely back to Earth (Lovell and Kluger, 2006 ). One of these challenges was controlling carbon dioxide levels onboard the space craft: “For 2 days straight [they] had worked on how to jury-rig the Odysseys canisters to the Aquarius's life support system. Now, using materials known to be available onboard the spacecraft—a sock, a plastic bag, the cover of a flight manual, lots of duct tape, and so on—the crew assembled a strange contraption and taped it into place. Carbon dioxide levels immediately began to fall into the safe range” (Team, 1970 ; Cass, 2005 ).

The success of Apollo 13's recovery from failure is often cited as a glowing example of human resourcefulness and inventiveness alongside more well-known inventions and innovations over the course of human history. However, this sort of inventive capability is not restricted to a few creative geniuses, but an ability present in all of us, and exemplified in the following mundane example. Consider a situation when your only suit is covered in lint and you do not own a lint remover. You see a roll of duct tape, and being resourceful you reason that it might be a good substitute. You then solve the problem of lint removal by peeling a full turn's worth of tape and re-attaching it backwards onto the roll to expose the sticky side all around the roll. By rolling it over your suit, you can now pick up all the lint.

In both these examples (historic as well as everyday), we see evidence for our innate ability to problem-solve in the real world. Solving real world problems in real time given constraints posed by one's environment are crucial for survival. At the core of this skill is our mental capability to get out of “sticky situations” or impasses, i.e., difficulties that appear unexpectedly as impassable roadblocks to solving the problem at hand. But, what are the cognitive processes that enable a problem solver to overcome such impasses and arrive at a solution, or at least a set of promising next steps?

A central aspect of this type of real world problem solving, is the role played by the solver's surrounding environment during the problem-solving process. Is it possible that interaction with one's environment can facilitate creative thinking? The answer to this question seems somewhat obvious when one considers the most famous anecdotal account of creative problem solving, namely that of Archimedes of Syracuse. During a bath, he found a novel way to check if the King's crown contained non-gold impurities. The story has traditionally been associated with the so-called “Eureka moment,” the sudden affective experience when a solution to a particularly thorny problem emerges. In this paper, I want to temporarily turn our attention away from the specific “aha!” experience itself and take particular note that Archimedes made this discovery, not with his eyes closed at a desk, but in a real-world context of a bath 1 . The bath was not only a passive, relaxing environment for Archimedes, but also a specific source of inspiration. Indeed it was his noticing the displacement of water that gave him a specific methodology for measuring the purity of the crown; by comparing how much water a solid gold bar of the same weight would displace as compared with the crown. This sort of continuous environmental interaction was present when the Apollo 13 engineers discovered their life-saving solution, and when you solved the suit-lint-removal problem with duct tape.

The neural mechanisms underlying problem-solving have been extensively studied in the literature, and there is general agreement about the key functional networks and nodes involved in various stages of problem-solving. In addition, there has been a great deal of work in studying the neural basis for creativity and insight problem solving, which is associated with the sudden emergence of solutions. However, in the context of problem-solving, creativity, and insight have been researched as largely an internal process without much interaction with and influence from the external environment (Wegbreit et al., 2012 ; Abraham, 2013 ; Kounios and Beeman, 2014 ) 2 . Thus, there are open questions of what role the environment plays during real world problem-solving (RWPS) and how the brain enables the assimilation of novel items during these external interactions.

In this paper, I synthesize the literature on problem-solving, creativity and insight, and particularly focus on how the environment can inform RWPS. I explore three environmentally-informed mechanisms that could play a critical role: (1) partial-cue driven context-shifting, (2) heuristic prototyping and learning novel associations, and (3) learning novel physical inferences. I begin first with some intuitions about real world problem solving, that might help ground this discussion and providing some key distinctions from more traditional problem solving research. Then, I turn to a review of the relevant literature on problem-solving, creativity, and insight first, before discussing the three above-mentioned environmentally-driven mechanisms. I conclude with a potential new model and map out its hypothesized neural basis.

2. Problem solving, creativity, and insight

2.1. what is real world problem-solving.

Archimedes was embodied in the real world when he found his solution. In fact, the real world helped him solve the problem. Whether or not these sorts of historic accounts of creative inspiration are accurate 3 , they do correlate with some of our own key intuitions about how problem solving occurs “in the wild.” Real world problem solving (RWPS) is different from those that occur in a classroom or in a laboratory during an experiment. They are often dynamic and discontinuous, accompanied by many starts and stops. Solvers are never working on just one problem. Instead, they are simultaneously juggling several problems of varying difficulties and alternating their attention between them. Real world problems are typically ill-defined, and even when they are well-defined, often have open-ended solutions. Coupled with that is the added aspect of uncertainty associated with the solver's problem solving strategies. As introduced earlier, an important dimension of RWPS is the continuous interaction between the solver and their environment. During these interactions, the solver might be inspired or arrive at an “aha!” moment. However, more often than not, the solver experiences dozens of minor discovery events— “hmmm, interesting…” or “wait, what?…” moments. Like discovery events, there's typically never one singular impasse or distraction event. The solver must iterate through the problem solving process experiencing and managing these sorts of intervening events (including impasses and discoveries). In summary, RWPS is quite messy and involves a tight interplay between problem solving, creativity, and insight. Next, I explore each of these processes in more detail and explicate a possible role of memory, attention, conflict management and perception.

2.2. Analytical problem-solving

In psychology and neuroscience, problem-solving broadly refers to the inferential steps taken by an agent 4 that leads from a given state of affairs to a desired goal state (Barbey and Barsalou, 2009 ). The agent does not immediately know how this goal can be reached and must perform some mental operations (i.e., thinking) to determine a solution (Duncker, 1945 ).

The problem solving literature divides problems based on clarity (well-defined vs. ill-defined) or on the underlying cognitive processes (analytical, memory retrieval, and insight) (Sprugnoli et al., 2017 ). While memory retrieval is an important process, I consider it as a sub-process to problem solving more generally. I first focus on analytical problem-solving process, which typically involves problem-representation and encoding, and the process of forming and executing a solution plan (Robertson, 2016 ).

2.2.1. Problem definition and representation

An important initial phase of problem-solving involves defining the problem and forming a representation in the working memory. During this phase, components of the prefrontal cortex (PFC), default mode network (DMN), and the dorsal anterior cingulate cortex (dACC) have been found to be activated. If the problem is familiar and well-structured, top-down executive control mechanisms are engaged and the left prefrontal cortex including the frontopolar, dorso-lateral (dlPFC), and ventro-lateral (vlPFC) are activated (Barbey and Barsalou, 2009 ). The DMN along with the various structures in the medial temporal lobe (MTL) including the hippocampus (HF), parahippocampal cortex, perirhinal and entorhinal cortices are also believed to have limited involvement, especially in episodic memory retrieval activities during this phase (Beaty et al., 2016 ). The problem representation requires encoding problem information for which certain visual and parietal areas are also involved, although the extent of their involvement is less clear (Anderson and Fincham, 2014 ; Anderson et al., 2014 ).

2.2.1.1. Working memory

An important aspect of problem representation is the engagement and use of working memory (WM). The WM allows for the maintenance of relevant problem information and description in the mind (Gazzaley and Nobre, 2012 ). Research has shown that WM tasks consistently recruit the dlPFC and left inferior frontal cortex (IC) for encoding an manipulating information; dACC for error detection and performance adjustment; and vlPFC and the anterior insula (AI) for retrieving, selecting information and inhibitory control (Chung and Weyandt, 2014 ; Fang et al., 2016 ).

2.2.1.2. Representation

While we generally have a sense for the brain regions that are functionally influential in problem definition, less is known about how exactly events are represented within these regions. One theory for how events are represented in the PFC is the structured event complex theory (SEC), in which components of the event knowledge are represented by increasingly higher-order convergence zones localized within the PFC, akin to the convergence zones (from posterior to anterior) that integrate sensory information in the brain (Barbey et al., 2009 ). Under this theory, different zones in the PFC (left vs. right, anterior vs. posterior, lateral vs. medial, and dorsal vs. ventral) represent different aspects of the information contained in the events (e.g., number of events to be integrated together, the complexity of the event, whether planning, and action is needed). Other studies have also suggested the CEN's role in tasks requiring cognitive flexibility, and functions to switch thinking modes, levels of abstraction of thought and consider multiple concepts simultaneously (Miyake et al., 2000 ).

Thus, when the problem is well-structured, problem representation is largely an executive control activity coordinated by the PFC in which problem information from memory populates WM in a potentially structured representation. Once the problem is defined and encoded, planning and execution of a solution can begin.

2.2.2. Planning

The central executive network (CEN), particularly the PFC, is largely involved in plan formation and in plan execution. Planning is the process of generating a strategy to advance from the current state to a goal state. This in turn involves retrieving a suitable solution strategy from memory and then coordinating its execution.

2.2.2.1. Plan formation

The dlPFC supports sequential planning and plan formation, which includes the generation of hypothesis and construction of plan steps (Barbey and Barsalou, 2009 ). Interestingly, the vlPFC and the angular gyrus (AG), implicated in a variety of functions including memory retrieval, are also involved in plan formation (Anderson et al., 2014 ). Indeed, the AG together with the regions in the MTL (including the HF) and several other regions form a what is known as the “core” network. The core network is believed to be activated when recalling past experiences, imagining fictitious, and future events and navigating large-scale spaces (Summerfield et al., 2010 ), all key functions for generating plan hypotheses. A recent study suggests that the AG is critical to both episodic simulation, representation, and episodic memory (Thakral et al., 2017 ). One possibility for how plans are formulated could involve a dynamic process of retrieving an optimal strategy from memory. Research has shown significant interaction between striatal and frontal regions (Scimeca and Badre, 2012 ; Horner et al., 2015 ). The striatum is believed to play a key role in declarative memory retrieval, and specifically helping retrieve optimal (or previously rewarded) memories (Scimeca and Badre, 2012 ). Relevant to planning and plan formation, Scimeca & Badre have suggested that the striatum plays two important roles: (1) in mapping acquired value/utility to action selection, and thereby helping plan formation, and (2) modulation and re-encoding of actions and other plan parameters. Different types of problems require different sets of specialized knowledge. For example, the knowledge needed to solve mathematical problems might be quite different (albeit overlapping) from the knowledge needed to select appropriate tools in the environment.

Thus far, I have discussed planning and problem representation as being domain-independent, which has allowed me to outline key areas of the PFC, MTL, and other regions relevant to all problem-solving. However, some types of problems require domain-specific knowledge for which other regions might need to be recruited. For example, when planning for tool-use, the superior parietal lobe (SPL), supramarginal gyrus (SMG), anterior inferior parietal lobe (AIPL), and certain portions of the temporal and occipital lobe involved in visual and spatial integration have been found to be recruited (Brandi et al., 2014 ). It is believed that domain-specific information stored in these regions is recovered and used for planning.

2.2.2.2. Plan execution

Once a solution plan has been recruited from memory and suitably tuned for the problem on hand, the left-rostral PFC, caudate nucleus (CN), and bilateral posterior parietal cortices (PPC) are responsible for translating the plan into executable form (Stocco et al., 2012 ). The PPC stores and maintains “mental template” of the executable form. Hemispherical division of labor is particularly relevant in planning where it was shown that when planning to solve a Tower of Hanoi (block moving) problem, the right PFC is involved in plan construction whereas the left PFC is involved in controlling processes necessary to supervise the execution of the plan (Newman and Green, 2015 ). On a separate note and not the focus of this paper, plan execution and problem-solving can require the recruitment of affective and motivational processing in order to supply the agent with the resolve to solve problems, and the vmPFC has been found to be involved in coordinating this process (Barbey and Barsalou, 2009 ).

2.3. Creativity

During the gestalt movement in the 1930s, Maier noted that “most instances of “real” problem solving involves creative thinking” (Maier, 1930 ). Maier performed several experiments to study mental fixation and insight problem solving. This close tie between insight and creativity continues to be a recurring theme, one that will be central to the current discussion. If creativity and insight are linked to RWPS as noted by Maier, then it is reasonable to turn to the creativity and insight literature for understanding the role played by the environment. A large portion of the creativity literature has focused on viewing creativity as an internal process, one in which the solvers attention is directed inwards, and toward internal stimuli, to facilitate the generation of novel ideas and associations in memory (Beaty et al., 2016 ). Focusing on imagination, a number of researchers have looked at blinking, eye fixation, closing eyes, and looking nowhere behavior and suggested that there is a shift of attention from external to internal stimuli during creative problem solving (Salvi and Bowden, 2016 ). The idea is that shutting down external stimuli reduces cognitive load and focuses attention internally. Other experiments studying sleep behavior have also noted the beneficial role of internal stimuli in problem solving. The notion of ideas popping into ones consciousness, suddenly, during a shower is highly intuitive for many and researchers have attempted to study this phenomena through the lens of incubation, and unconscious thought that is internally-driven. There have been several theories and counter-theories proposed to account specifically for the cognitive processes underlying incubation (Ritter and Dijksterhuis, 2014 ; Gilhooly, 2016 ), but none of these theories specifically address the role of the external environment.

The neuroscience of creativity has also been extensively studied and I do not focus on an exhaustive literature review in this paper (a nice review can be found in Sawyer, 2011 ). From a problem-solving perspective, it has been found that unlike well-structured problems, ill-structured problems activate the right dlPFC. Most of the past work on creativity and creative problem-solving has focused on exploring memory structures and performing internally-directed searches. Creative idea generation has primarily been viewed as internally directed attention (Jauk et al., 2012 ; Benedek et al., 2016 ) and a primary mechanism involved is divergent thinking , which is the ability to produce a variety of responses in a given situation (Guilford, 1962 ). Divergent thinking is generally thought to involve interactions between the DMN, CEN, and the salience network (Yoruk and Runco, 2014 ; Heinonen et al., 2016 ). One psychological model of creative cognition is the Geneplore model that considers two major phases of generation (memory retrieval and mental synthesis) and exploration (conceptual interpretation and functional inference) (Finke et al., 1992 ; Boccia et al., 2015 ). It has been suggested that the associative mode of processing to generate new creative association is supported by the DMN, which includes the medial PFC, posterior cingulate cortex (PCC), tempororparietal juntion (TPJ), MTL, and IPC (Beaty et al., 2014 , 2016 ).

That said, the creativity literature is not completely devoid of acknowledging the role of the environment. In fact, it is quite the opposite. Researchers have looked closely at the role played by externally provided hints from the time of the early gestalt psychologists and through to present day studies (Öllinger et al., 2017 ). In addition to studying how hints can help problem solving, researchers have also looked at how directed action can influence subsequent problem solving—e.g., swinging arms prior to solving the two-string puzzle, which requires swinging the string (Thomas and Lleras, 2009 ). There have also been numerous studies looking at how certain external perceptual cues are correlated with creativity measures. Vohs et al. suggested that untidiness in the environment and the increased number of potential distractions helps with creativity (Vohs et al., 2013 ). Certain colors such as blue have been shown to help with creativity and attention to detail (Mehta and Zhu, 2009 ). Even environmental illumination, or lack thereof, have been shown to promote creativity (Steidle and Werth, 2013 ). However, it is important to note that while these and the substantial body of similar literature show the relationship of the environment to creative problem solving, they do not specifically account for the cognitive processes underlying the RWPS when external stimuli are received.

2.4. Insight problem solving

Analytical problem solving is believed to involve deliberate and conscious processing that advances step by step, allowing solvers to be able to explain exactly how they solved it. Inability to solve these problems is often associated with lack of required prior knowledge, which if provided, immediately makes the solution tractable. Insight, on the other hand, is believed to involve a sudden and unexpected emergence of an obvious solution or strategy sometimes accompanied by an affective aha! experience. Solvers find it difficult to consciously explain how they generated a solution in a sequential manner. That said, research has shown that having an aha! moment is neither necessary nor sufficient to insight and vice versa (Danek et al., 2016 ). Generally, it is believed that insight solvers acquire a full and deep understanding of the problem when they have solved it (Chu and Macgregor, 2011 ). There has been an active debate in the problem solving community about whether insight is something special. Some have argued that it is not, and that there are no special or spontaneous processes, but simply a good old-fashioned search of a large problem space (Kaplan and Simon, 1990 ; MacGregor et al., 2001 ; Ash and Wiley, 2006 ; Fleck, 2008 ). Others have argued that insight is special and suggested that it is likely a different process (Duncker, 1945 ; Metcalfe, 1986 ; Kounios and Beeman, 2014 ). This debate lead to two theories for insight problem solving. MacGregor et al. proposed the Criterion for Satisfactory Progress Theory (CSPT), which is based on Newell and Simons original notion of problem solving as being a heuristic search through the problem space (MacGregor et al., 2001 ). The key aspect of CSPT is that the solver is continually monitoring their progress with some set of criteria. Impasses arise when there is a criterion failure, at which point the solver tries non-maximal but promising states. The representational change theory (RCT) proposed by Ohlsson et al., on the other hand, suggests that impasses occur when the goal state is not reachable from an initial problem representation (which may have been generated through unconscious spreading activation) (Ohlsson, 1992 ). In order to overcome an impasse, the solver needs to restructure the problem representation, which they can do by (1) elaboration (noticing new features of a problem), (2) re-encoding fixing mistaken or incomplete representations of the problem, and by (3) changing constraints. Changing constraints is believed to involve two sub-processes of constraint relaxation and chunk-decomposition.

The current position is that these two theories do not compete with each other, but instead complement each other by addressing different stages of problem solving: pre- and post-impasse. Along these lines, Ollinger et al. proposed an extended RCT (eRCT) in which revising the search space and using heuristics was suggested as being a dynamic and iterative and recursive process that involves repeated instances of search, impasse and representational change (Öllinger et al., 2014 , 2017 ). Under this theory, a solver first forms a problem representation and begins searching for solutions, presumably using analytical problem solving processes as described earlier. When a solution cannot be found, the solver encounters an impasse, at which point the solver must restructure or change the problem representation and once again search for a solution. The model combines both analytical problem solving (through heuristic searches, hill climbing and progress monitoring), and creative mechanisms of constraint relaxation and chunk decomposition to enable restructuring.

Ollingers model appears to comprehensively account for both analytical and insight problem solving and, therefore, could be a strong candidate to model RWPS. However, while compelling, it is nevertheless an insufficient model of RWPS for many reasons, of which two are particularly significant for the current paper. First, the model does explicitly address mechanisms by which external stimuli might be assimilated. Second, the model is not sufficiently flexible to account for other events (beyond impasse) occurring during problem solving, such as distraction, mind-wandering and the like.

So, where does this leave us? I have shown the interplay between problem solving, creativity and insight. In particular, using Ollinger's proposal, I have suggested (maybe not quite explicitly up until now) that RWPS involves some degree of analytical problem solving as well as the post-impasse more creative modes of problem restructuring. I have also suggested that this model might need to be extended for RWPS along two dimensions. First, events such as impasses might just be an instance of a larger class of events that intervene during problem solving. Thus, there needs to be an accounting of the cognitive mechanisms that are potentially influenced by impasses and these other intervening events. It is possible that these sorts of events are crucial and trigger a switch in attentional focus, which in turn facilitates switching between different problem solving modes. Second, we need to consider when and how externally-triggered stimuli from the solver's environment can influence the problem solving process. I detail three different mechanisms by which external knowledge might influence problem solving. I address each of these ideas in more detail in the next two sections.

3. Event-triggered mode switching during problem-solving

3.1. impasse.

When solving certain types of problems, the agent might encounter an impasse, i.e., some block in its ability to solve the problem (Sprugnoli et al., 2017 ). The impasse may arise because the problem may have been ill-defined to begin with causing incomplete and unduly constrained representations to have been formed. Alternatively, impasses can occur when suitable solution strategies cannot be retrieved from memory or fail on execution. In certain instances, the solution strategies may not exist and may need to be generated from scratch. Regardless of the reason, an impasse is an interruption in the problem solving process; one that was running conflict-free up until the point when a seemingly unresolvable issue or an error in the predicted solution path was encountered. Seen as a conflict encountered in the problem-solving process it activates the anterior cingulate cortex (ACC). It is believed that the ACC not only helps detect the conflict, but also switch modes from one of “exploitation” (planning) to “exploration” (search) (Quilodran et al., 2008 ; Tang et al., 2012 ), and monitors progress during resolution (Chu and Macgregor, 2011 ). Some mode switching duties are also found to be shared with the AI (the ACC's partner in the salience network), however, it is unclear exactly the extent of this function-sharing.

Even though it is debatable if impasses are a necessary component of insight, they are still important as they provide a starting point for the creativity (Sprugnoli et al., 2017 ). Indeed, it is possible that around the moment of impasse, the AI and ACC together, as part of the salience network play a crucial role in switching thought modes from analytical planning mode to creative search and discovery mode. In the latter mode, various creative mechanisms might be activated allowing for a solution plan to emerge. Sowden et al. and many others have suggested that the salience network is potentially a candidate neurobiological mechanism for shifting between thinking processes, more generally (Sowden et al., 2015 ). When discussing various dual-process models as they relate to creative cognition, Sowden et al. have even noted that the ACC activation could be useful marker to identify shifting as participants work creative problems.

3.2. Defocused attention

As noted earlier, in the presence of an impasse there is a shift from an exploitative (analytical) thinking mode to an exploratory (creative) thinking mode. This shift impacts several networks including, for example, the attention network. It is believed attention can switch between a focused mode and a defocused mode. Focused attention facilitates analytic thought by constraining activation such that items are considered in a compact form that is amenable to complex mental operations. In the defocused mode, agents expand their attention allowing new associations to be considered. Sowden et al. ( 2015 ) note that the mechanism responsible for adjustments in cognitive control may be linked to the mechanisms responsible for attentional focus. The generally agreed position is that during generative thinking, unconscious cognitive processes activated through defocused attention are more prevalent, whereas during exploratory thinking, controlled cognition activated by focused attention becomes more prevalent (Kaufman, 2011 ; Sowden et al., 2015 ).

Defocused attention allows agents to not only process different aspects of a situation, but to also activate additional neural structures in long term memory and find new associations (Mendelsohn, 1976 ; Yoruk and Runco, 2014 ). It is believed that cognitive material attended to and cued by positive affective state results in defocused attention, allowing for more complex cognitive contexts and therefore a greater range of interpretation and integration of information (Isen et al., 1987 ). High attentional levels are commonly considered a typical feature of highly creative subjects (Sprugnoli et al., 2017 ).

4. Role of the environment

In much of the past work the focus has been on treating creativity as largely an internal process engaging the DMN to assist in making novel connections in memory. The suggestion has been that “individual needs to suppress external stimuli and concentrate on the inner creative process during idea generation” (Heinonen et al., 2016 ). These ideas can then function as seeds for testing and problem-solving. While true of many creative acts, this characterization does not capture how creative ideas arise in many real-world creative problems. In these types of problems, the agent is functioning and interacting with its environment before, during and after problem-solving. It is natural then to expect that stimuli from the environment might play a role in problem-solving. More specifically, it can be expected that through passive and active involvement with the environment, the agent is (1) able to trigger an unrelated, but potentially useful memory relevant for problem-solving, (2) make novel connections between two events in memory with the environmental cue serving as the missing link, and (3) incorporate a completely novel information from events occuring in the environment directly into the problem-solving process. I explore potential neural mechanisms for these three types of environmentally informed creative cognition, which I hypothesize are enabled by defocused attention.

4.1. Partial cues trigger relevant memories through context-shifting

I have previously discussed the interaction between the MTL and PFC in helping select task-relevant and critical memories for problem-solving. It is well-known that pattern completion is an important function of the MTL and one that enables memory retrieval. Complementary Learning Theory (CLS) and its recently updated version suggest that the MTL and related structures support initial storage as well as retrieval of item and context-specific information (Kumaran et al., 2016 ). According to CLS theory, the dentate gyrus (DG) and the CA3 regions of the HF are critical to selecting neural activity patterns that correspond to particular experiences (Kumaran et al., 2016 ). These patterns might be distinct even if experiences are similar and are stabilized through increases in connection strengths between the DG and CA3. Crucially, because of the connection strengths, reactivation of part of the pattern can activate the rest of it (i.e., pattern completion). Kumaran et al. have further noted that if consistent with existing knowledge, these new experiences can be quickly replayed and interleaved into structured representations that form part of the semantic memory.

Cues in the environment provided by these experiences hold partial information about past stimuli or events and this partial information converges in the MTL. CLS accounts for how these cues might serve to reactivate partial patterns, thereby triggering pattern completion. When attention is defocused I hypothesize that (1) previously unnoticed partial cues are considered, and (2) previously noticed partial cues are decomposed to produce previously unnoticed sub-cues, which in turn are considered. Zabelina et al. ( 2016 ) have shown that real-world creativity and creative achievement is associated with “leaky attention,” i.e., attention that allows for irrelevant information to be noticed. In two experiments they systematically explored the relationship between two notions of creativity— divergent thinking and real-world creative achievement—and the use of attention. They found that attentional use is associated in different ways for each of the two notions of creativity. While divergent thinking was associated with flexible attention, it does not appear to be leaky. Instead, selective focus and inhibition components of attention were likely facilitating successful performance on divergent thinking tasks. On the other hand, real-world creative achievement was linked to leaky attention. RWPS involves elements of both divergent thinking and of real-world creative achievement, thus I would expect some amount of attentional leaks to be part of the problem solving process.

Thus, it might be the case that a new set of cues or sub-cues “leak” in and activate memories that may not have been previously considered. These cues serve to reactivate a diverse set of patterns that then enable accessing a wide range of memories. Some of these memories are extra-contextual, in that they consider the newly noticed cues in several contexts. For example, when unable to find a screwdriver, we might consider using a coin. It is possible that defocused attention allows us to consider the coin's edge as being a potentially relevant cue that triggers uses for the thin edge outside of its current context in a coin. The new cues (or contexts) may allow new associations to emerge with cues stored in memory, which can occur during incubation. Objects and contexts are integrated into memory automatically into a blended representation and changing contexts disrupts this recognition (Hayes et al., 2007 ; Gabora, 2016 ). Cue-triggered context shifting allows an agent to break-apart a memory representation, which can then facilitate problem-solving in new ways.

4.2. Heuristic prototyping facilitates novel associations

It has long been the case that many scientific innovations have been inspired by events in nature and the surrounding environment. As noted earlier, Archimedes realized the relationship between the volume of an irregularly shaped object and the volume of water it displaced. This is an example of heuristic prototyping where the problem-solver notices an event in the environment, which then triggers the automatic activation of a heuristic prototype and the formation of novel associations (between the function of the prototype and the problem) which they can then use to solve the problem (Luo et al., 2013 ). Although still in its relative infancy, there has been some recent research into the neural basis for heuristic prototyping. Heuristic prototype has generally been defined as an enlightening prototype event with a similar element to the current problem and is often composed of a feature and a function (Hao et al., 2013 ). For example, in designing a faster and more efficient submarine hull, a heuristic prototype might be a shark's skin, while an unrelated prototype might be a fisheye camera (Dandan et al., 2013 ).

Research has shown that activating the feature function of the right heuristic prototype and linking it by way of semantic similarity to the required function of the problem was the key mechanism people used to solve several scienitific insight problems (Yang et al., 2016 ). A key region activated during heuristic prototyping is the dlPFC and it is believed to be generally responsible for encoding the events into memory and may play an important role in selecting and retrieving the matched unsolved technical problem from memory (Dandan et al., 2013 ). It is also believed that the precuneus plays a role in automatic retrieval of heuristic information allowing the heuristic prototype and the problem to combine (Luo et al., 2013 ). In addition to semantic processing, certain aspects of visual imagery have also been implicated in heuristic prototyping leading to the suggestion of the involvement of Broadman's area BA 19 in the occipital cortex.

There is some degree of overlap between the notions of heuristic prototyping and analogical transfer (the mapping of relations from one domain to another). Analogical transfer is believed to activate regions in the left medial fronto-parietal system (dlPFC and the PPC) (Barbey and Barsalou, 2009 ). I suggest here that analogical reasoning is largely an internally-guided process that is aided by heuristic prototyping which is an externally-guided process. One possible way this could work is if heuristic prototyping mechanisms help locate the relevant memory with which to then subsequently analogize.

4.3. Making physical inferences to acquire novel information

The agent might also be able to learn novel facts about their environment through passive observation as well as active experimentation. There has been some research into the neural basis for causal reasoning (Barbey and Barsalou, 2009 ; Operskalski and Barbey, 2016 ), but beyond its generally distributed nature, we do not know too much more. Beyond abstract causal reasoning, some studies looked into the cortical regions that are activated when people watch and predict physical events unfolding in real-time and in the real-world (Fischer et al., 2016 ). It was found that certain regions were associated with representing types of physical concepts, with the left intraparietal sulcus (IPS) and left middle frontal gyrus (MFG) shown to play a role in attributing causality when viewing colliding objects (Mason and Just, 2013 ). The parahippocampus (PHC) was associated with linking causal theory to observed data and the TPJ was involved in visualizing movement of objects and actions in space (Mason and Just, 2013 ).

5. Proposed theory

I noted earlier that Ollinger's model for insight problem solving, while serving as a good candidate for RWPS, requires extension. In this section, I propose a candidate model that includes some necessary extensions to Ollinger's framework. I begin by laying out some preliminary notions that underlie the proposed model.

5.1. Dual attentional modes

I propose that the attention-switching mechanism described earlier is at the heart of RWPS and enables two modes of operation: focused and defocused mode. In the focused mode, the problem representation is more or less fixed, and problem solving proceeds in a focused and goal directed manner through search, planning, and execution mechanisms. In the defocused mode, problem solving is not necessarily goal directed, but attempts to generate ideas, driven by both internal and external items.

At first glance, these modes might seem similar to convergent and divergent thinking modes postulated by numerous others to account for creative problem solving. Divergent thinking allows for the generation of new ideas and convergent thinking allows for verification and selection of generated ideas. So, it might seem that focused mode and convergent thinking are similar and likewise divergent and defocused mode. They are, however, quite different. The modes relate less to idea generation and verification, and more to the specific mechanisms that are operating with regard to a particular problem at a particular moment in time. Convergent and divergent processes may be occurring during both defocused and focused modes. Some degree of divergent processes may be used to search and identify specific solution strategies in focused mode. Also, there might be some degree of convergent idea verification occuring in defocused mode as candidate items are evaluated for their fit with the problem and goal. Thus, convergent and divergent thinking are one amongst many mechanisms that are utilized in focused and defocused mode. Each of these two modes has to do with degree of attention placed on a particular problem.

There have been numerous dual-process and dual-systems models of cognition proposed over the years. To address criticisms raised against these models and to unify some of the terminology, Evans & Stanovich proposed a dual-process model comprising Type 1 and Type 2 thought (Evans and Stanovich, 2013 ; Sowden et al., 2015 ). Type 1 processes are those that are believed to be autonomous and do not require working memory. Type 2 processes, on the other hand, are believed to require working memory and are cognitively decoupled to prevent real-world representations from becoming confused with mental simulations (Sowden et al., 2015 ). While acknowledging various other attributes that are often used to describe dual process models (e.g., fast/slow, associative/rule-based, automatic/controlled), Evans & Stanovich note that these attributes are merely frequent correlates and not defining characteristics of Type 1 or Type 2 processes. The proposed dual attentional modes share some similarities with the Evans & Stanovich Type 1 and 2 models. Specifically, Type 2 processes might occur in focused attentional mode in the proposed model as they typically involve the working memory and certain amount of analytical thought and planning. Similarly, Type 1 processes are likely engaged in defocused attentional mode as there are notions of associative and generative thinking that might be facilitated when attention has been defocused. The crucial difference between the proposed model and other dual-process models is that the dividing line between focused and defocused attentional modes is the degree of openness to internal and external stimuli (by various networks and functional units in the brain) when problem solving. Many dual process models were designed to classify the “type” of thinking process or a form of cognitive processing. In some sense, the “processes” in dual process theories are characterized by the type of mechanism of operation or the type of output they produced. Here, I instead characterize and differentiate the modes of thinking by the receptivity of different functional units in the brain to input during problem solving.

This, however, raises a different question of the relationship between these attentional modes and conscious vs. unconscious thinking. It is clear that both the conscious and unconscious are involved in problem solving, as well as in RWPS. Here, I claim that a problem being handled is, at any given point in time, in either a focused mode or in a defocused mode. When in the focused mode, problem solving primarily proceeds in a manner that is available for conscious deliberation. More specifically, problem space elements and representations are tightly managed and plans and strategies are available in the working memory and consciously accessible. There are, however, secondary unconscious operations in the focused modes that includes targeted memory retrieval and heuristic-based searches. In the defocused mode, the problem is primarily managed in an unconscious way. The problem space elements are broken apart and loosely managed by various mechanisms that do not allow for conscious deliberation. That said, it is possible that some problem parameters remain accessible. For example, it is possible that certain goal information is still maintained consciously. It is also possible that indexes to all the problems being considered by the solver are maintained and available to conscious awareness.

5.2. RWPS model

Returning to Ollinger's model for insight problem solving, it now becomes readily apparent how this model can be modified to incorporate environmental effects as well as generalizing the notion of intervening events beyond that of impasses. I propose a theory for RWPS that begins with standard analytical problem-solving process (See Figures ​ Figures1, 1 , ​ ,2 2 ).

Summary of neural activations during focused problem-solving (Left) and defocused problem-solving (Right) . During defocused problem-solving, the salience network (insula and ACC) coordinates the switching of several networks into a defocused attention mode that permits the reception of a more varied set of stimuli and interpretations via both the internally-guided networks (default mode network DMN) and externally guided networks (Attention). PFC, prefrontal cortex; ACC, anterior cingulate cortex; PCC, posterior cingulate cortex; IPC, inferior parietal cortex; PPC, posterior parietal cortex; IPS, intra-parietal sulcus; TPJ, temporoparietal junction; MTL, medial temporal lobe; FEF, frontal eye field.

Proposed Model for Real World Problem Solving (RWPS). The corresponding neural correlates are shown in italics. During problem-solving, an initial problem representation is formed based on prior knowledge and available perceptual information. The problem-solving then proceeds in a focused, goal-directed mode until the goal is achieved or a defocusing event (e.g., impasse or distraction) occurs. During focused mode operation, the solver interacts with the environment in directed manner, executing focused plans, and allowing for predicted items to be activated by the environment. When a defocusing event occurs, the problem-solving then switches into a defocused mode until a focusing event (e.g., discovery) occurs. In defocused mode, the solver performs actions unrelated to the problem (or is inactive) and is receptive to a set of environmental triggers that activate novel aspects using the three mechanisms discussed in this paper. When a focusing event occurs, the diffused problem elements cohere into a restructured representation and problem-solving returns into a focused mode.

5.2.1. Focused problem solving mode

Initially, both prior knowledge and perceptual entities help guide the creation of problem representations in working memory. Prior optimal or rewarding solution strategies are obtained from LTM and encoded in the working memory as well. This process is largely analytical and the solver interacts with their environment through focused plan or idea execution, targeted observation of prescribed entities, and estimating prediction error of these known entities. More specifically, when a problem is presented, the problem representations are activated and populated into working memory in the PFC, possibly in structured representations along convergence zones. The PFC along with the Striatum and the MTL together attempt at retrieving an optimal or previously rewarded solution strategy from long term memory. If successfully retrieved, the solution strategy is encoded into the PPC as a mental template, which then guides relevant motor control regions to execute the plan.

5.2.2. Defocusing event-triggered mode switching

The search and solve strategy then proceeds analytically until a “defocusing event” is encountered. The salience network (AI and ACC) monitor for conflicts and attempt to detect any such events in the problem-solving process. As long as no conflicts are detected, the salience network focuses on recruiting networks to achieve goals and suppresses the DMN (Beaty et al., 2016 ). If the plan execution or retrieval of the solution strategy fails, then a defocusing event is detected and the salience network performs mode switching. The salience network dynamically switches from the focused problem-solving mode to a defocused problem-solving mode (Menon, 2015 ). Ollinger's current model does not account for other defocusing events beyond an impasse, but it is not inconceivable that there could be other such events triggered by external stimuli (e.g., distraction or an affective event) or by internal stimuli (e.g., mind wandering).

5.2.3. Defocused problem solving mode

In defocused mode, the problem is operated on by mechanisms that allow for the generation and testing of novel ideas. Several large-scale brain networks are recruited to explore and generate new ideas. The search for novel ideas is facilitated by generally defocused attention, which in turn allows for creative idea generation from both internal as well as external sources. The salience network switches operations from defocused event detection to focused event or discovery detection, whereby for example, environmental events or ideas that are deemed interesting can be detected. During this idea exploration phase, internally, the DMN is no longer suppressed and attempts to generate new ideas for problem-solving. It is known that the IPC is involved in the generation of new ideas (Benedek et al., 2014 ) and together with the PPC in coupling different information together (Simone Sandkühler, 2008 ; Stocco et al., 2012 ). Beaty et al. ( 2016 ) have proposed that even this internal idea-generation process can be goal directed, thereby allowing for a closer working relationship between the CEN and the DMN. They point to neuroimaging evidence that support the possibility that the executive control network (comprising the lateral prefrontal and inferior parietal regions) can constrain and direct the DMN in its process of generating ideas to meet task-specific goals via top down monitoring and executive control (Beaty et al., 2016 ). The control network is believed to maintain an “internal train of thought” by keeping the task goal activated, thereby allowing for strategic and goal-congruent searches for ideas. Moreover, they suggest that the extent of CEN involvement in the DMN idea-generation may depend on the extent to which the creative task is constrained. In the RWPS setting, I would suspect that the internal search for creative solutions is not entirely unconstrained, even in the defocused mode. Instead, the solver is working on a specified problem and thus, must maintain the problem-thread while searching for solutions. Moreover, self-generated ideas must be evaluated against the problem parameters and thereby might need some top-down processing. This would suggest that in such circumstances, we would expect to see an increased involvement of the CEN in constraining the DMN.

On the external front, several mechanisms are operating in this defocused mode. Of particular note are the dorsal attention network, composed of the visual cortex (V), IPS and the frontal eye field (FEF) along with the precuneus and the caudate nucleus allow for partial cues to be considered. The MTL receives synthesized cue and contextual information and populates the WM in the PFC with a potentially expanded set of information that might be relevant for problem-solving. The precuneus, dlPFC and PPC together trigger the activation and use of a heuristic prototype based on an event in the environment. The caudate nucleus facilitates information routing between the PFC and PPC and is involved in learning and skill acquisition.

5.2.4. Focusing event-triggered mode switching

The problem's life in this defocused mode continues until a focusing event occurs, which could be triggered by either external (e.g., notification of impending deadline, discovery of a novel property in the environment) or internal items (e.g., goal completion, discovery of novel association or updated relevancy of a previously irrelevant item). As noted earlier, an internal train of thought may be maintained that facilitates top-down evaluation of ideas and tracking of these triggers (Beaty et al., 2016 ). The salience network switches various networks back to the focused problem-solving mode, but not without the potential for problem restructuring. As noted earlier, problem space elements are maintained somewhat loosely in the defocused mode. Thus, upon a focusing event, a set or subset of these elements cohere into a tight (restructured) representation suitable for focused mode problem solving. The process then repeats itself until the goal has been achieved.

5.3. Model predictions

5.3.1. single-mode operation.

The proposed RWPS model provides several interesting hypotheses, which I discuss next. First, the model assumes that any given problem being worked on is in one mode or another, but not both. Thus, the model predicts that there cannot be focused plan execution on a problem that is in defocused mode. The corollary prediction is that novel perceptual cues (as those discussed in section 4) cannot help the solver when in focused mode. The corollary prediction, presumably has some support from the inattentional blindness literature. Inattentional blindness is when perceptual cues are not noticed during a task (e.g., counting the number of basketball passes between several people, but not noticing a gorilla in the scene) (Simons and Chabris, 1999 ). It is possible that during focused problem solving, that external and internally generated novel ideas are simply not considered for problem solving. I am not claiming that these perceptual cues are always ignored, but that they are not considered within the problem. Sometimes external cues (like distracting occurrences) can serve as defocusing events, but the model predicts that the actual content of these cues are not themselves useful for solving the specific problem at hand.

When comparing dual-process models Sowden et al. ( 2015 ) discuss shifting from one type of thinking to another and explore how this shift relates to creativity. In this regard, they weigh the pros and cons of serial vs. parallel shifts. In dual-process models that suggest serial shifts, it is necessary to disengage one type of thought prior to engaging the other or to shift along a continuum. Whereas, in models that suggest parallel shifts, each of the thinking types can operate in parallel. Per this construction, the proposed RWPS model is serial, however, not quite in the same sense. As noted earlier, the RWPS model is not a dual-process model in the same sense as other dual process model. Instead, here, the thrust is on when the brain is receptive or not receptive to certain kinds of internal and external stimuli that can influence problem solving. Thus, while the modes may be serial with respect to a certain problem, it does not preclude the possibility of serial and parallel thinking processes that might be involved within these modes.

5.3.2. Event-driven transitions

The model requires an event (defocusing or focusing) to transition from one mode to another. After all why else would a problem that is successfully being resolved in the focused mode (toward completion) need to necessarily be transferred to defocused mode? These events are interpreted as conflicts in the brain and therefore the mode-switching is enabled by the saliency network and the ACC. Thus, the model predicts that there can be no transition from one mode to another without an event. This is a bit circular, as an event is really what triggers the transition in the first place. But, here I am suggesting that an external or internal cue triggered event is what drives the transition, and that transitions cannot happen organically without such an event. In some sense, the argument is that the transition is discontinuous, rather than a smooth one. Mind-wandering is good example of when we might drift into defocused mode, which I suggest is an example of an internally driven event caused by an alternative thought that takes attention away from the problem.

A model assumption underlying RWPS is that events such as impasses have a similar effect to other events such as distraction or mind wandering. Thus, it is crucial to be able to establish that there exists of class of such events and they have a shared effect on RWPS, which is to switch attentional modes.

5.3.3. Focused mode completion

The model also predicts that problems cannot be solved (i.e., completed) within the defocused mode. A problem can be considered solved when a goal is reached. However, if a goal is reached and a problem is completed in the defocused mode, then there must have not been any converging event or coherence of problem elements. While it is possible that the solver arbitrarily arrived at the goal in a diffused problem space and without conscious awareness of completing the task or even any converging event or problem recompiling, it appears somewhat unlikely. It is true that there are many tasks that we complete without actively thinking about it. We do not think about what foot to place in front of another while walking, but this is not an instance of problem solving. Instead, this is an instance of unconscious task completion.

5.3.4. Restructuring required

The model predicts that a problem cannot return to a focused mode without some amount of restructuring. That is, once defocused, the problem is essentially never the same again. The problem elements begin interacting with other internally and externally-generated items, which in turn become absorbed into the problem representation. This prediction can potentially be tested by establishing some preliminary knowledge, and then showing one group of subjects the same knowledge as before, while showing the another group of subjects different stimuli. If the model's predictions hold, the problem representation will be restructured in some way for both groups.

There are numerous other such predictions, which are beyond the scope of this paper. One of the biggest challenges then becomes evaluating the model to set up suitable experiments aimed at testing the predictions and falsifying the theory, which I address next.

6. Experimental challenges and paradigms

One of challenges in evaluating the RWPS is that real world factors cannot realistically be accounted for and sufficiently controlled within a laboratory environment. So, how can one controllably test the various predictions and model assumptions of “real world” problem solving, especially given that by definition RWPS involves the external environment and unconscious processing? At the expense of ecological validity, much of insight problem solving research has employed an experimental paradigm that involves providing participants single instances of suitably difficult problems as stimuli and observing various physiological, neurological and behavioral measures. In addition, through verbal protocols, experimenters have been able to capture subjective accounts and problem solving processes that are available to the participants' conscious. These experiments have been made more sophisticated through the use of timed-hints and/or distractions. One challenge with this paradigm has been the selection of a suitable set of appropriately difficult problems. The classic insight problems (e.g., Nine-dot, eight-coin) can be quite difficult, requiring complicated problem solving processes, and also might not generalize to other problems or real world problems. Some in the insight research community have moved in the direction of verbal tasks (e.g., riddles, anagrams, matchstick rebus, remote associates tasks, and compound remote associates tasks). Unfortunately, these puzzles, while providing a great degree of controllability and repeatability, are even less realistic. These problems are not entirely congruent with the kinds of problems that humans are solving every day.

The other challenge with insight experiments is the selection of appropriate performance and process tracking measures. Most commonly, insight researchers use measures such as time to solution, probability of finding solution, and the like for performance measures. For process tracking, verbal protocols, coded solution attempts, and eye tracking are increasingly common. In neuroscientific studies of insight various neurological measures using functional magnetic resonance imaging (fMRI), electroencephalography (EEGs), transcranial direct current stimulation (tDCS), and transcranial magnetic stimulation (tMS) are popular and allow for spatially and temporally localizing an insight event.

Thus, the challenge for RWPS is two-fold: (1) selection of stimuli (real world problems) that are generalizable, and (2) selection of measures (or a set of measures) that can capture key aspects of the problem solving process. Unfortunately, these two challenges are somewhat at odds with each other. While fMRI and various neuroscientific measures can capture the problem solving process in real time, it is practically difficult to provide participants a realistic scenario while they are laying flat on their back in an fMRI machine and allowed to move nothing more than a finger. To begin addressing this conundrum, I suggest returning to object manipulation problems (not all that different from those originally introduced by Maier and Duncker nearly a century ago), but using modern computing and user-interface technologies.

One pseudo-realistic approach is to generate challenging object manipulation problems in Virtual Reality (VR). VR has been used to describe 3-D environment displays that allows participants to interact with artificially projected, but experientially realistic scenarios. It has been suggested that virtual environments (VE) invoke the same cognitive modules as real equivalent environmental experience (Foreman, 2010 ). Crucially, since VE's can be scaled and designed as desired, they provide a unique opportunity to study pseudo-RWPS. However, a VR-based research approach has its limitations, one of which is that it is nearly impossible to track participant progress through a virtual problem using popular neuroscientific measures such as fMRI because of the limited mobility of connected participants.

Most of the studies cited in this paper utilized an fMRI-based approach in conjunction with a verbal or visual task involving problem-solving or creative thinking. Very few, if any, studies involved the use physical manipulation, and those physical manipulations were restricted to limited finger movements. Thus, another pseudo-realistic approach is allowing subjects to teleoperate robotic arms and legs from inside the fMRI machine. This paradigm has seen limited usage in psychology and robotics, in studies focused on Human-Robot interaction (Loth et al., 2015 ). It could be an invaluable tool in studying real-time dynamic problem-solving through the control of a robotic arm. In this paradigm a problem solving task involving physical manipulation is presented to the subject via the cameras of a robot. The subject (in an fMRI) can push buttons to operate the robot and interact with its environment. While the subjects are not themselves moving, they can still manipulate objects in the real world. What makes this paradigm all the more interesting is that the subject's manipulation-capabilities can be systematically controlled. Thus, for a particular problem, different robotic perceptual and manipulation capabilities can be exposed, allowing researchers to study solver-problem dynamics in a new way. For example, even simple manipulation problems (e.g., re-arranging and stacking blocks on a table) can be turned into challenging problems when the robotic movements are restricted. Here, the problem space restrictions are imposed not necessarily on the underlying problem, but on the solver's own capabilities. Problems of this nature, given their simple structure, may enable studying everyday practical creativity without the burden of devising complex creative puzzles. Crucial to note, both these pseudo-realistic paradigms proposed demonstrate a tight interplay between the solver's own capabilities and their environment.

7. Conclusion

While the neural basis for problem-solving, creativity and insight have been studied extensively in the past, there is still a lack of understanding of the role of the environment in informing the problem-solving process. Current research has primarily focused on internally-guided mental processes for idea generation and evaluation. However, the type of real world problem-solving (RWPS) that is often considered a hallmark of human intelligence has involved both a dynamic interaction with the environment and the ability to handle intervening and interrupting events. In this paper, I have attempted to synthesize the literature into a unified theory of RWPS, with a specific focus on ways in which the environment can help problem-solve and the key neural networks involved in processing and utilizing relevant and useful environmental information. Understanding the neural basis for RWPS will allow us to be better situated to solve difficult problems. Moreover, for researchers in computer science and artificial intelligence, clues into the neural underpinnings of the computations taking place during creative RWPS, can inform the design the next generation of helper and exploration robots which need these capabilities in order to be resourceful and resilient in the open-world.

Author contributions

The author confirms being the sole contributor of this work and approved it for publication.

Conflict of interest statement

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

I am indebted to Professor Matthias Scheutz, Professor Elizabeth Race, Professor Ayanna Thomas, and Professor. Shaun Patel for providing guidance with the research and the manuscript. I am also grateful for the facilities provided by Tufts University, Medford, MA, USA.

1 My intention is not to ignore the benefits of a concentrated internal thought process which likely occurred as well, but merely to acknowledge the possibility that the environment might have also helped.

2 The research in insight does extensively use “hints” which are, arguably, a form of external influence. But these hints are highly targeted and might not be available in this explicit form when solving problems in the real world.

3 The accuracy of these accounts has been placed in doubt. They often are recounted years later, with inaccuracies, and embellished for dramatic effect.

4 I use the term “agent” to refer to the problem-solver. The term agent is more general than “creature” or “person” or “you" and is intentionally selected to broadly reference humans, animals as well as artificial agents. I also selectively use the term “solver.”

Funding. The research for this Hypothesis/Theory Article was funded by the authors private means. Publication costs will be covered by my institution: Tufts University, Medford, MA, USA.

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