The Character of a Problem-Solver

 The hallmark of every great problem-solver is fidelity to facts and reason.

1. Introduction.

The character of a great problem-solver is an assemblage of assorted properties that often seem contradictory. How can you be analytical and yet open to subjective thinking? How is it realistic to be both flexible and decisive? The best problem solvers prefer a balance of characteristics. Too much of one doesn’t expiate another. All are needed, but in what measure for what problem? For the NASA Mars landing, analytics skills dominate. Maintaining diplomacy with an enemy requires all of them. To coach a major sports team, one needs analysis, flexibility, discipline, and great knowledge of the game and team. Yet the coach must be aware of the impact of all decisions and of their biases. Coaching has strategic and tactical components, together with the analytic and subjective.

Rare is the real-world problem for which only a single answer is correct, absolute, incontrovertible, and required. Therefore, the modern problem-solver must begin with a foundation of given evidence and facts, rules of inference, solution criteria, and, if possible, optimal solution properties and a stopping time. The problem is thus stated or framed. The solution process begins. The nature of the process also has a set of taboos that must not contravene rules for determining the solution. For example, the detective simply cannot recommend indictment of the suspect because he/she knows him to be guilty. Problem-solving is not for absolutists, who are given to finding absolutes, where none may exist. Nor can the cleric disprove a miracle because there is no proof that God exists. Negotiating a labor contract is more than quid pro quo analysis. It involves the (subjective) moods of management and equally the resolve of labor. Rules of inference, i.e. the logic du jour, are significant. This is Section 2.

All this sounds interesting. The beginner may want only the pointers with a promise to follow them. Not so. There are pitfalls to all. They tend to deal with the human capacity to make errors, and worse to make errors of which they are unaware. Section 3 addresses poor leadership for big problems, and Section 4 discusses the manifold of human subjective issues we have with information, data, and knowledge, and how they can be mentally corrupted  

2. Characteristics of the Problem-Solver.

The qualities, skills, aptitudes, whatever they are called, for the skilled problem-solver are as varied as they are remarkable. Most of the descriptors in the given list (Figure 1) are ranked as admirable by all people everywhere. They celebrate the best in us; for problems, they define us. Even more remarkably, they apply to humanity’s greatest challenges, those of solving diverse and often impossibly difficult problems. That is what we do. No longer tool makers of note, humankind is now problem-solvers par excellence. This implies being prepared, making tools when needed, making theories, explaining mysteries, and conquering uncertainty. We have overtaken all preceding industrial revolutions, even though we created them. Currently, we live in awe and fear of the next revolution, Artificial Intelligence (AI). Yet AI is not a problem-solving tool. It can produce all manner of devices and even appears enlightened, but we can be deceived by what it can do. At this point, it cannot match us, in inspiration, innovation, and intuition, nor theory and creativity. Below are identified eleven abilities or characteristics of problem-solvers, slowly learned over generations of thought. When all are combined, powerful forces are revealed. As usual, we note all the cited abilities are interdependent. Also, we emphasize these problems are not of the school’s variety of single steps, such as applying a formula, recalling facts, making a paragraph, or thinking of something clever.  These are serious problems of magnitude and scale.

While the style of our explanations suggests teams at work, many apply to just the single person working a problem, not simple ones from grade school, perhaps, but of life, as in professional careers, international relations, family and parenting, and spiritual matters. Good parents, for example, are either good problem-solvers or just plain lucky. As you plunge into this collection, note that each, in its way, is a marker of human intelligence. Any person scoring well on all has clearly achieved wisdom.

1.     Analytical – The great problem-solver possesses strong analytical skills, can break down complex problems into components, see patterns, pursue likelihoods, and recognize cause-and-effect relationships. In that light, the great problem-solver well understands that correlation is not causality. Preferring objective thinking, they are open to subjective factors, a delicate balance to maintain.  Overall, they tend to be pragmatic by nature and totally consumed by logic[1]. The analytical thinker is comfortable with data and theory, can distinguish conjecture from fact, and avoids being swept up in emotional clouds. Yet, even they are not immune to subjective possibilities and indeterminate hypotheses. See Section 4.

2.     Knowledge and Experience – The great problem-solver knows what the problem is about and has vast knowledge about it. One cannot solve a problem with little knowledge about it, no matter how bright one is. The goal is always to penetrate the problem for understanding, and particularly to know what others have done to solve similar problems. Approaching the problem tabula rasa rarely works. The second component revealed is experience. If it is agreed that knowledge is critical, then even more so is experience in solving problems. Experience implies the solver has an organizational framework for approaching a problem, decomposing it into relevant components, managing teams, and guiding progress. Experience is indispensable. Most books on problem-solving emphasize experience. For most senior-level jobs, experience is often the determining criterion for hiring.

 

3.      

Figure 1 Characteristics of the Solver

 

4.     Creative – The creative talent that all problem-solvers need is perhaps the most difficult, if not elusive, to achieve. Though very important, its meaning is vague, and the results can be questionable. Similar to innovation and inspiration, creativity is an important part of all phases of solving a problem. However creative, as in it’s new, is different from creative, as in it helps. It forms a bridge between sections of the solution and even helps in the decomposition of the problem into just the right components. Sometimes creative thinking is called lateral thinking, and is likened to “thinking outside the box.” However, the great problem-solver knows full well what’s inside the box, i.e. knows their stuff, before looking afar.

 5.     Persistent – Persistence, like perseverance, is essential. Its opposite, that of just giving up, is the bane of problem-solving, yet after weeks, it’s so easy to cave in. Targets must be assigned. Frustration must be avoided. Make clear estimates on when to wrap it up and give the best solution you have. Pushing any particular method to some conclusion is required[2], but still pivoting to alternatives must always be on the table. 

 6.     Tenacious – Here we include companion characteristics such as discipline, singlemindedness, focus, aggressiveness, hardheadedness, perspicacity, shrewdness, and many more often called unsavory qualities. Tenaciousness is essential for all big problems, complex problems, and those intractable, impossible problems. Tenacity alone is full-time work. Yet tenacity can manifest in rigidity, stubbornness, and pugnacity. Tenacity, therefore, is another of those qualities requiring a delicate balance.

 

7.     Resilience – Part of solving big problems is failing; a corresponding part is recovery. Consider the relevant maxim, “Many of life's failures are people who did not realize how close they were to success when they gave up.” —Thomas Edison. This is resilience[3]. Every parent knows that resilience is necessary to do the job properly. Every diplomat understands alternatives will be offered, demanded, and must stand on alert to remain on track. Good parents are also good diplomats. Another part of resilience is navigating changes in the problem landscape, managing difficult partners, maintaining composure, handling setbacks, keeping an open dialogue, and seeking alternative pathways. Therefore, part of resilience is maintaining a positive attitude, strong motivation, and managing stress. Abraham Lincoln, the most resilient of Presidents, in the prosecution of the American Civil War was only to be matched a few decades later by the resolve of Winston Churchill during World War II. History was written by sheer resilience in the face of adversity. We would be remiss not to mention countless scientists who spent years solving important problems, facing down failure after failure.

 

 8.     Constant learner – You know that the more you know, the more problems you can solve. The great problem-solver maintains a constant interest in knowing more about everything. This can be challenging because often new learning is not on task, but it broadens your perspective, your toolkit, your grasp of diverse topics, and especially your catalog of problems with solutions. You always have some article or book at hand. You enjoy learning new skills, which is a critical asset and a companion to experience.  For example, the current hot topic of Artificial Intelligence (AI) is job-threatening to almost everyone. Knowing about AI and how to use it helps you remain above the fray, toward retooling yourself with skills that keep you a step ahead of the grim AI reaper. You are learning to use AI to help you solve problems it cannot. More traditionally, what will you do if your boss asks one day if you’ve tried this or that? You need to be robust in logging your failed attempts. It is not easy to document failure. Knowing and remembering what doesn’t work is almost always helpful.

9.     Flexible – Flexibility is the beginning of wisdom when it comes to solving problems. The solver who takes a fixed attitude on how a problem should be solved has lost the game from the onset. Seeking objective certitude should not be the solution plan for real-world problems. If the solution comes on your first attempt, the problem is too easy – not that easy is bad. It’s just unusual. Educators grapple with instructing widely diverse students' attitudes and abilities, but must herd their classroom of “cats” to the common goal of learning the topic. If this doesn’t take flexibility, I’m not sure what does.

 

10.  Open-minded – Being unbiased, willing to listen, willing to explore, and willing to take risks are all important. “Risk” is an interesting self-reflective word. Taking a risk is risky. Taking risks essentially implies uncertainty. As well, fighting bias in risk is itself a constant problem. Bias wants a certain type of solution and steers you there. Fighting internal conflicts of bias is continual. The great problem-solver does not object to experts in other areas considering the problem, mostly because they usually bring expertise “outside the box” to bear. This does prove out on occasion. In Section 4, we discuss what bias can do, especially when you think you don’t have it.

11.  Collaborative – It helps to be talkative, and even to talk to yourself (Illeism[4]). When talking, you are also listening. You not only work with your team, analyzing, considering, dreaming, conjecturing, and guessing, but you also communicate with others, usually on specific points, especially when the big problem is too large to explain. You can and do consider diverse opinions. Think of your many channels of communication: talking, listening, seeing/viewing, reading, interactivity, thinking, and talking to yourself. While all may seem similar, they challenge different parts of the brain. In solving big problems, it’s all hands on deck. Keep notes. The crazy suggestion yesterday may be the key idea tomorrow.  The familiar maxim, “From the mouths of babes comes …,” does apply - sometimes.

Figure 2.  Overly Complex Solution

Another aspect of collaboration is the communication of the solution. Eventually, having solved the big problem, you must communicate the solution to the stakeholders. This point is simply expressed in the extreme.

If a solution is found that no one understands, is it still a solution?

The solution may still need work. Budgetary, diplomatic, materials, management, policy, logistical, and other factors enter. For the solution: Have unnecessary, non-evidentiary, or hypothetical assumptions been made? At times, we make assumptions just to get a handle on the solution. Later, we whittle those unnecessary assumptions down as much as possible. However, hypothetical assumptions can lead to incorrect solutions, and hopefully, they will be removed. For example, assuming in a data security problem that “all people are honest” is completely hypothetical, and therefore if “such and such” is our solution it is probably dead wrong. Have principles of parsimony been applied to minimize monetary or other tangible factors? Have principles of simplicity been applied to render the solution more comprehensible? Is there adherence to Einstein’s razor to “make the solution as simple as possible, but no simpler?”  Getting too simple leads to analogy, a wonderful tool for offering an intuitive glimpse, but usually unactionable for implementation.

In the academy, the most common situation is that the problem is worked on to become publishable, and then it is written and shipped for publication. When first sending astronauts to space, NASA expended millions to develop a “space pen” that could function in the vacuum and zero gravity of space. This pen is still available to purchase. The Russians, having a smaller budget, decided to use pencils. Parsimony!

12.  Decisive – The act of decisiveness is both critical and delicate.  For example, when should the problem-solving stop? The great problem-solver can assess options, evaluate risks and benefits, and make precise decisions within deadlines. In big problems, degrees of solutions are always available. Is the current quality of the solution sufficient?  One temptation is to declare victory over the problem when more work should be done. Decisiveness can function as a substitute for close-mindedness or simple exhaustion. Thoroughness must accompany decisiveness. Another component of decisiveness is to introduce review mechanisms and peer evaluations to evaluate problem-solving processes and decisions.  Remarkably, decisiveness requires controlled patience. Another interesting aspect of (competent) decisiveness is to come to no conclusion or solution but to leave the problem open. This signals farther reaching work must be done. The more ideological team or dogmatic group might fall victim to the expression, “Relying on ideology is the same as saying, "I know, therefore I don't have to think."

This point is important. Suppose you and/or your team have a solution at hand, or think you have. Now is the time to ask whether there is a simpler, more efficient, and cheaper solution. This is Occam’s razor.

3. The Problem-Solving Paradox.

In this brief section, we’ll look at some big problems, those requiring a team or entire division to solve. For these, a capable leader is essential. The list of character qualities is detailed above. These people are no lightweights in problem-solving. They are innovative, creative, and decisive. The paradox is that for local, state, and federal governments they are more frequently than not, little more than political appointees. Their problem-solving expertise is almost null.  In years past, many came from the business world and had long experience with solving problems. However, recently, many, too many, have been chosen from political activist posts where their only ability is to think in ideological terms, that is, in terms of political solutions.

Our example is to consider the core list of problems faced by every large city. Certainly, there are others.

•                 Poverty: Large cities often have high rates of poverty, leading to a lack of access to basics such as housing, food, water, and healthcare.

•                 Crime: High rates of crime make it dangerous for people to live, work, and play in the city.

•                 Traffic congestion: Congested traffic can make it difficult for people to get around and often leads to air pollution.

•                 Pollution: Large cities often have high levels of pollution caused by traffic, factories, and other sources, and this, in turn, leads to health problems.

•                 Housing: Large cities often have a shortage of affordable housing. This can make it difficult for people to find a place to live and can lead to homelessness.

•                 Education: Large cities often have underfunded schools. This can lead to a lack of quality education for children living in the city.

•                 Healthcare: Large cities often have underfunded hospitals and clinics, implying a diminished quality of healthcare.

•                 Transportation: Large cities often have a poor public transportation system.

•                 Infrastructure: Large cities often have aging infrastructure, a chronic problem everywhere. It’s too easy to rob infrastructure budgets for other purposes.

•                 Governance: Large cities are often difficult to govern a result of which includes. corruption, bureaucracy, and gridlock. City councils jam through legislation that makes little practical sense, preferring to rule on what should be as opposed to what is.

Imagine the problems of leading a recovery from any of these problems. Most would require years of study, but appointees are too often unmotivated and incapable, and temporary. The result is that the problems, managed by equally unmotivated staff, get worse. Governments have similar, and often more difficult problems. It is no wonder big city problems never seem to be solved. The problem-solving paradox is everywhere, particularly in huge enterprises such as corporations and universities.

The only supersized agency with well-studied leaders is the military. They have studied war and the military machine at the highest levels. Many are scholars of their craft. Their biggest shortcomings seem to be with fiscal management, and the happenstance when the government interferes internally in military policy. Having an $800 billion budget, the military, together with its War College, should have a companion Finance College.

4. What Can Go Wrong?

For solving problems, we know bad logic, false assumptions or theories (discussed in an earlier chapter), bad leadership, shortness of time, lack of information, exhaustion, insufficient preparation, bias, and lack of thoroughness can cause bad outcomes. These are merely the obvious issues. More subtle conditions, events, and actions are equally common. They function like parasites of thought, and specifically for problem-solving. What’s worse, these parasites are universal, having infected us all.

Imagination can be a problem, for obvious reasons. Let me give a personal example. We’ve read about the Big Bang theory of the origin of the universe; we read about 95% of the universe is composed of dark matter and dark energy, meaning it is undetectable other than its gravitational effect, and that is necessary to keep the universe intact. So, putting these together, I hypothesized (imagined) that dark matter/energy is residual from a previous universe. So, is it? For this, I used ChatGPT[5]. It turns out that this is but one of the many guesses about the origin of the universe. No evidence or any observation supports it, but there it is. Just another guess. This illustrates why knowledge is important for problem-solving; amateur guesses are fine but contribute little.

The most obvious aspect of what can go wrong is subjectivity, which, in a philosophical context, is the view that truth depends on an individual's conscious experience. It refers to how someone's judgment is shaped by personal opinions and feelings that interfere with outside influences. The most common is bias, the bane of all problem-solvers. To some extent, they amplify and clarify earlier in this chapter. We postpone subjectivity for a few paragraphs in favor of other, lesser-known factors.

Besides logical fallacies, other problems arise in constructing solutions.

Counterfactual thinking[6] and reasoning argue from facts such as events and reasons based on causation that had certain conditions been changed, the outcome would have been different. These are “What if …?” and “If only …” type questions. Such questions argue that by changing initial conditions, it is possible to change the outcome. For example, Alice studies for one hour for an exam and gets a “C” grade. She concludes that had she studied three hours, she would have a grade of “A.” Of course, Alice simply may not know how to rigorously study for three hours, or whether she just cannot understand the materials, no matter how many hours are given. These just begin to count the alternate possibilities.

The sports team manager is facing a losing season and reasons that if only he had signed the budding star, his team would have fared better. Thus, he now goes to find a budding star, expecting a winning future next year. The problem here is that he cannot know that just that one single factor was the cause of a losing season. Apparently, sports teams have learned all about counterfactual thinking and now track players and team performance with data-driven analytics.

Counterfactual thinking allows individuals to mentally explore and evaluate different possibilities. It can help understand or explore causality. It can be viewed as similar to thought experiments. But the derived causality can be as specious as correct. The serious problem-solver must always be on alert for counterfactual thinking, compelling solutions or conjectures, though every one of us falls for it at one time or another.

Hyperintensional[7] Reasoning is a concept that explores the complexities of intentional[8] action. Challenging traditional notions of intentionality, it suggests that our intentions often corrupt or surpass our conscious awareness, thereby extending into implicit and unconscious mental states. Hyperintensional thinking

·       Can focus on different attitudes or attitudes toward propositions, such as the distinction between belief, disbelief, information, counterfactual conditionals, and suspension of judgment.

·       Implies our mental states are influenced by implicit biases, automatic associations, and unconscious processes that steer our intentions and behaviors.

·       Recognizes the significant role of how unconscious processes shape our intentions.

Specifically, our responses (via True or False) can be different in assessing two necessarily equivalent propositions. Applied to problem-solving, hyperintensionalism suggests that our intentions and thought processes extend beyond conscious awareness, and this influences how we approach problems, how we understand them, and how we solve them. One test of susceptibility to hyperintensionalism is to rephrase a statement in two quite different ways and consider how you respond. (It’s better to get someone else to do the rephrasing.) Clearly, this must be included as one source of misunderstandings. See below. Basically, our mental states, emotions, and biases affect our problem-solving abilities. Understanding hyperintensional thinking, naturally occurring in us all, makes us aware of how we can better account for flawed analysis and exploration, concomitant with the complexities inherent in language, thought, and reasoning, in offering our solutions.

Note the similarities to framing, discussed next.

Framing is a relatively new discovery in psychology, though it’s been intuitively understood for millennia. Roughly, how the problem is framed has a profound effect on how it is received, with a positive frame preferred over the negative. Formulated in 1981 by Tversky and Kahneman,[9] it seems to be endemic in our culture. For example, if one says the chance of success in getting the job is 40% and the chance of failure is 60%, logically equivalent, people are more likely to select the more positive statement. Here are a few more examples.

·       While looking for a disinfectant, you choose a product that claims to kill 95% of all the germs over one that claims that 5% of the germs will survive.

·       A politician might frame a policy as "protecting the environment" instead of "raising taxes,” though both imply taxes will be raised.

·       The government may suggest we go to war to save our allies in another country, rather than give estimates of lives lost or money expended.

While not so much in scientific disciplines, framing plays a large role in more social problems, particularly if decision-makers are susceptible. Kahneman and Tversky also formulated prospect theory[10], a psychological theory of choice. The upshot is that people do not behave as rational actors.

Gettier thinking is the devil incarnate. Returning to our example of building a beltway, we drop in on the land acquisition officer. Her job is to acquire the hundreds of land parcels on which to construct the highway. She selected a typical landowner, offered to take them to dinner, and related the grand plan. Toward the end of the dinner, she suggested a price the Beltway committee would pay. He immediately accepted. Delighted, she reported the next day to explain how to get all the land. She had a formula. Easy, yes? What she did not know is that the landowner was happy to dump his property. As well, she did not know that her method was based on the false assumption that “schmoozing” with landowners was the method to use.

This is an example of being correct in buying the land but for false reasons. And this is Gettier thinking[11]. A simpler example is from the office. Joe has determined that someone in the office has a Chevy. He makes his conclusion by talking to Bill, who does not have a Chevy and doesn’t know anyone who does. Yet, he is correct because someone in the office actually does have a Chevy. Again, we have a correct conclusion derived from false information. Let’s did a little deeper. Joe believes his conclusion; his conclusion is true; he has justified his conclusion. This type of example, in 1963, shook the concept of knowledge and epistemology. Gettier problems are quite a hot topic in philosophical circles today, having shaken the long-standing belief that justified true belief implies knowledge.

Similar to Gettier thinking, false assumptions almost always lead to poor or questionable solutions. They can lead to false simplifications of the entire problem-solving enterprise and then to quick solutions. When decisions are made upon such solutions, the flaws are sometimes determined, often at a great cost in treasure or lives. Decisive team leaders or just the single problem-solver must always become the grand inquisitor about the validity of what assumptions have been made.

Incomplete or inaccurate models are common for impossible or complex problems. For both it is sometimes impossible to model all the factors and variables in the problem, the result being the model is incomplete and therefore inaccurate. Even when the number of factors or variables is small, it is easy to make an inaccurate model to solve the problem. For example, this author modeled the best possible (future) Marathon speeds using five different models[12]. Five different best speeds were determined, but which, if any, are true? This is a super-serious problem in predicting the future when there are no precedent examples to check predictions against.

Misconceptions and misunderstandings can be the ruination of even young students in math class. If you don’t understand the problem, how can you solve it? Remember a formula wrong is common. For complex problems, the difficulty is exponential. The team or just the single problem-solver must be certain that there are no problem misconceptions or misunderstandings in the formulation of the problem and in the steps in solving it. These were considered in detail in an earlier chapter.

Preassigned conclusions are a consequence of assuming what the solution should be and then expending all efforts to prove it. This leads to logical fallacies and other false reasoning effects.

Subjectivity is the final topic in this section. Let’s first compare this topic with rationality. Some value may be uncovered by considering, by way of analogy, that problem-solving has two gods, rationality and subjectivity, the first being akin to a god of fertility, giving life and the forces of nature, while the second is a mischievous trickster, capable of shapeshifting, unnaturally influencing the natural world, and responsible for disease, death, and drought[13]. This comparison may seem a bit far afield, but sometimes it helps when trying to understand the difference between two opposing forces and how they function.

Subjectivity is not all bad. It is responsible for taste in art, music, food, and all manner of preferences. However, bias and ignorance, central to this discussion, are sometimes disguised as righteousness or absoluteness. About problems, especially difficult problems where there is little certainty, subjectivity can be destructive.  Numerous ways explain how this happens, though we’ll consider only a few.

·       Impact bias, also known as the "duration bias" or the "emotional forecasting error," is a cognitive bias that refers to the tendency to under- or overestimate the impact that future events will have on our emotional reactions. This can lead to making poor decisions or even taking risks. One reason is to focus on the immediacy of the event while deprecating future impact.

·       Limited perspective is the signature of the incompetent problem-solver. Not only are they unable to see beyond their local sandbox, they simply don’t want to. This may not be to say “ignorance is bliss,” but more to say the solver is content with their body of knowledge and problem-solving techniques. Limited perspective is often a consequence of age, but all too often, younger people seem to enter the world with an idee fixe. Rigidity should not be a quality of youth.

·       Also associated with personal problem-solving incompetence is the Dunning-Kreuger effect[14], for which there is a vast literature. It is another cognitive bias where a low ability is manifest in the overestimation of competence. Conversely, those with high abilities tend to underestimate their abilities. There is a host of other conditions involving self-estimates of abilities, including the superiority complex, self-deception, the overconfidence effect, and self-serving bias.

·       Inefficient analysis of the problem due to subjectivity can lead to missing root causes of the true problem revealing analyses that are incomplete or in error. They lead to serious blunders in every sort of problem.

·       Resistance to change is also an emotional factor that causes the problem-solver to refuse to consider new solutions beyond their knowledge or beliefs. The refusal to adapt to new information is an earmark of resistance.  

·       Lack of transparency can be attributed to an emotional reaction to potential difficulties with a pending decision. It is also a consequence of unfairness and revealing methods of decision. Frequently, associated problems with a lack of transparency require even more concealment of facts and results. Sustaining such transparency issues often require media support, or worse, ultimate executive authority.

·       Consensus difficulty is always a problem when subjective factors in the form of strong opinions or biased leadership are present. It seems certain that every reader has experienced strong opinions. This can also lead to internal conflicts if not downright hostility. These, in turn, can lead to selecting the wrong solution.

·       Inaccuracy can also be subjective. It can be caused by improperly reading the situation clearly and therefore making decision errors. In engineering, this can result in catastrophic errors. For more social problems, it may lead to exacerbating the problem and making it worse. Inaccuracy is also not necessarily the subjective result of simply failing to consider critical factors in the problem equation. A classic example is the loss of the Challenger spacecraft, where engineers did not consider the cracking effect of freezing temperatures on the O-ring rubber-like seals separating rocket booster segments. The Nobel laureate Richard Feynman, a problem-solver par excellence, did[15]. The disaster was explained using a glass of freezing water and a specimen of the O-ring – before the cameras, no less. Not too bad for a physicist who specialized in quantum electrodynamics, just a bit off-topic.

Subjectivity, logic, and language, in their many forms, constitute only three factors causing difficulties in understanding and solving problems. The problems implicit in problem-solving above tap into mental issues difficult to understand and pinpoint. Almost all are deniable, and moreover, almost all are unrecognized even when evidence is offered.  

5. Conclusions.

The effective problem-solver is much like a Swiss Army Knife, with an assortment of skills and tools ready for any occasion. As emphasized in Section 2, experience is essential, together with many other factors, not the least of which is decisiveness is far more subtle than simply “calling the shots.” General intelligence, though implied, is implicit in its entirety, as highlighted in Section 3. What the great problem-solvers of math competitions have in common is that they have practiced ad nauseam. What helps as a life’s exercise is to view every situation from a problem-to-solution perspective.  This helps you understand the conclusions presented, but even more, it helps to expose weaknesses in what you are expected to believe. Never, ever underestimate the power of practice[16]. Another method employed is learning how to explain an idea or method with just the right balance of words and technical detail. Even the structure of your explanations is important, as it reveals how you think.

In Section 4, we considered a number of the lesser-known pitfalls of problem-solving, many of which can be traced to bias, but others to more subtle aspects of logical flaws, false assumptions, and language. Interestingly, on the scene, relative newcomers to psychology and philosophy have been expressed. To know them all is to be aware of a few assorted and diverse traps.

Problem-solvers are focused, even super-focused. In fact, some have trouble just relaxing. Getting that way (i.e. with a supercharged focus) requires lots of practice by solving lots of problems, dedicated reading, and hours upon hours of hard work. Importantly, problem-solvers love to focus, use their minds, solve problems, and take great pleasure in having solved them[17]. Focused problem-solvers are always on high alert, ready to pounce on any fresh idea.

A final note, though it sounds a bit preachy, is that schools and colleges should emphasize problem-solving to graduate better problem-solvers. For example, high school teachers, instead of asking students to write a theme about how they spent their summer vacation, could ask, “Write a theme on the problems you had this summer and how you solved them.”  College calculus teachers could ask, “For a given cost function, compare the derivative with marginal cost.” Nonstandard? Yes. But it forces students to think objectively and to solve a problem.

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20. Masten, A. S. (2001). Ordinary magic: Resilience processes in development. American Psychologist, 56(3), 227–238.
21. Southwick, S. M., & Charney, D. S. (2012). Resilience: The science of mastering life’s greatest challenges. Cambridge University Press.
22. Holling, C. S. (1973). Resilience and stability of ecological systems. Annual Review of Ecology and Systematics, 4, 1–23.

23. Kolb, D. A. (1984). Experiential learning: Experience as the source of learning and development. Prentice Hall.
24. Argyris, C., & Schön, D. A. (1978). Organizational learning: A theory of action perspective. Addison-Wesley.

25. Holland, J. H. (1992). Adaptation in natural and artificial systems. MIT Press.
26. Tetlock, P. E., & Gardner, D. (2015). Superforecasting: The art and science of prediction. Crown.

27. Woolley, A. W., Chabris, C. F., Pentland, A., Hashmi, N., & Malone, T. W. (2010). Evidence for a collective intelligence factor in the performance of human groups. Science, 330(6004), 686–688.
28. Hackman, J. R. (2002). Leading teams: Setting the stage for great performances. Harvard Business School Press.

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[1] It may be of some value to consult, William James, The Collected Works of William James, wherein he discusses at length about finding truth. Direct comparisons are available comparing truth with determination of problem solutions. Additionally, logic is no longer Aristotelian; modal logics are an important factor.

[2] The world record for persistence, if records are kept, must go to Thomas A. Edison(1847-1931)  who tried at least 6,000 filament types to find one that glowed well and lasted long. As well, Louis Pasteur (1822-1895) spent unlimited hours developing a cure for rabies, the scourge of diseases of all time.

[3] Resilience is one of the most celebrated words in language that implies the ability of a system to withstand changes in its environment and still function.

[4] https://www.linkedin.com/posts/g-donald-allen-420b0315_what-is-illeism-both-ancient-and-new-we-activity-7055966739100037120-qpR_?utm_source=share&utm_medium=member_desktop

[5] Large Language Models (LLM), such as chatGPT, are now a part of our tools for discovery, at least for what is known. They are not great for fine detail or deep research, but tremendously helpful for general information. All LLMs are not the same. Bard said no, there is no evidence without commentary about it being a hypothesis.

[6] Roese, N. (1997). "Counterfactual thinking". Psychological Bulletin. 121 (1): 133–148.

[7] Nolan, Daniel. Intensionality and hyperintensionality (2019) doi:10.4324/9780415249126-

DD3603-1. Routledge Encyclopedia of Philosophy, Taylor and Francis,

https://www.rep.routledge.com/articles/thematic/intensionality-and-hyperintensionality/v-1.

Copyright © 1998-2023 Routledge.

[8] To clarify the words, “intentional” refers to an aim or purpose while “intensional” with an “s” refers to meaning.

[9] Tversky, A., & Kahneman, D. (1981). The framing of decisions and the psychology of choice. science, 211(4481), 453-458.

[10] Kahneman, D., & Tversky, A. (2013). Prospect theory: An analysis of decision under risk . In  Handbook of the fundamentals of financial decision making: Part I 

[11] Gettier, Edmund L. (1 June 1963). "Is Justified True Belief Knowledge?". Analysis. 23 (6): 121–123.

[12]  Allen, G. Donald, (2018), Modeling World Record Predictions in Track Events IOSR Journal of Sports and Physical Education (IOSR-JSPE) e-ISSN: 2347-6737, p-ISSN: 2347-6745, Volume 5, Issue 6, (Nov. – Dec. 2018), PP 06-22.

[13] This handy analogy is based on two gods, within San spirituality, of the African Kalihari Bushmen, /Xam and Mantis (aka /Kaggen), having similar roles, one good and the other bad. Precise details are somewhat murky, considering most traditions and religion of the Kalihari Bushmen have historically only been transmitted orally.

[14] Dunning, David (2011). "Chapter Five – The Dunning–Kruger Effect: On Being Ignorant of One's Own Ignorance". Advances in Experimental Social Psychology. Vol. 44. Academic Press. pp. 247–296.

[15] Online at http://www.feynman.com/science/the-challenger-disaster/

[16] Suppose you need some serious surgery. Who do you want holding the scalpel, a brilliant young doctor just out of med school or an experienced surgeon with a record of dozens of successful similar interventions? You don’t deny the young surgeon should have a chance to operate, but just not on you. Bias or caution?

[17] Feynman, Richard P. (1999). Robbins, Jeffrey (ed.). The Pleasure of Finding Things Out: The Best Short Works of Richard P. Feynman. Cambridge, Massachusetts: Perseus Books.

© 2026 G. Donald Allen
Professor of Mathematics, Emeritus
Texas A&M University
College Station, TX 77843
gdonaldallen@gmail.com