Problem Solving

2. DEVISING A PLAN

2. DEVISING A PLAN • Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution. • Have you seen it before? Or have you seen the same problem in a slightly different form? • Do you know a related problem? Do you know a theorem that could be useful? • Look at the unknown! Try to think of a familiar problem having the same or a similar unknown. • Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible? • Could you restate the problem? Could you restate it still differently? Go back to definitions. • If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other? • Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

Decision Tree

A decision tree is a decision support tool that uses a tree-like graph or model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm. Decision trees are commonly used in operations research, specifically in decision analysis, to help identify a strategy most likely to reach a goal, but are also a popular tool in machine learning.

Metaphor

A metaphor is a figure of speech that refers, for rhetorical effect, to one thing by mentioning another thing.[1] It may provide clarity or identify hidden similarities between two ideas. Where a simile compares two items, a metaphor directly equates them, and does not use "like" or "as" as does a simile. One of the most commonly cited examples of a metaphor in English literature is the "All the world's a stage" monologue from As You Like It: All the world's a stage, And all the men and women merely players; They have their exits and their entrances[...] —William Shakespeare, As You Like It, 2/7[2]

Abstraction

Abstraction in its main sense is a conceptual process by which general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abstraction" is the product of this process — a concept that acts as a super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category.

Common barriers to problem solving

Common barriers to problem solving are mental constructs that impede our ability to correctly solve problems. These barriers prevent people from solving problems in the most efficient manner possible. Five of the most common processes and factors that researchers have identified as barriers to problem solving are confirmation bias, mental set, functional fixedness, unnecessary constraints, and irrelevant information.

Creative Computing

Creative computing covers the area of creativity and computing, addressing the issue of knowledge discovery.

Creative Problem Solving

Creative problem-solving, a type of problem solving, is the mental process of searching for a new and novel creative solution to a problem, a solution which is novel, original and not obvious. To qualify as creative problem-solving, the solution must solve the stated problem in a novel way, and the solution must be reached independently.[1] Creative problem-solving usually begins with defining the problem. This may lead to a simple non-creative solution, or to finding a "textbook solution". The creative problem-solving process may also lead to the discovery of prior art and of creative solutions by others. The process, in these cases, may then be abandoned, if the discovered solution is "good enough".[2][3] Typically a creative solution will have 'elegant' characteristics such as using existing components without introducing any new components into the solution (i.e. "no moving parts"), using the problematic factor as the basis for the solution (i.e. "using the Philistine's sword against him"), or involving a change of perspective (i.e. the line through nine dots riddle).[4] Many times a solution is considered creative if components that are readily available can be used, and when there is a short time limit within which to solve the problem.[5] These two factors are typical to the solutions shown in the MacGyver (TV series).

Creativity

Creativity is a phenomenon whereby something new and somehow valuable is formed. The created item may be intangible (such as an idea, a scientific theory, a musical composition, or a joke) or a physical object (such as an invention, a literary work, or a painting). Scholarly interest in creativity involves many definitions and concepts pertaining to a number of disciplines: psychology, cognitive science, education, philosophy (particularly philosophy of science), technology, theology, sociology, linguistics, business studies, songwriting, and economics, covering the relations between creativity and general intelligence, mental and neurological processes, personality type and creative ability, creativity and mental health; the potential for fostering creativity through education and training, especially as augmented by technology; and the application of creative resources to improve the effectiveness of teaching and learning.

Creativity techniques

Creativity techniques are methods that encourage creative actions, whether in the arts or sciences. They focus on a variety of aspects of creativity, including techniques for idea generation and divergent thinking, methods of re-framing problems, changes in the affective environment and so on. They can be used as part of problem solving, artistic expression, or therapy. Some techniques require groups of two or more people while other techniques can be accomplished alone. These methods include word games, written exercises and different types of improvisation, or algorithms for approaching problems. Aleatory techniques exploiting randomness are also common.

Critical Thinking Procedure

Critical thinking calls for the ability to: Recognize problems, to find workable means for meeting those problems Understand the importance of prioritization and order of precedence in problem solving Gather and marshal pertinent (relevant) information Recognize unstated assumptions and values Comprehend and use language with accuracy, clarity, and discernment Interpret data, to appraise evidence and evaluate arguments Recognize the existence (or non-existence) of logical relationships between propositions Draw warranted conclusions and generalizations Put to test the conclusions and generalizations at which one arrives Reconstruct one's patterns of beliefs on the basis of wider experience Render accurate judgments about specific things and qualities in everyday life In sum: "A persistent effort to examine any belief or supposed form of knowledge in the light of the evidence that supports or refutes it and the further conclusions to which it tends."[13]

Third principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work, discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals. 3. CARRYING OUT THE PLAN • Third. Carry out your plan. • Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

First principle: Understand the problem

"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as: What are you asked to find or show? Can you restate the problem in your own words? Can you think of a picture or a diagram that might help you understand the problem? Is there enough information to enable you to find a solution? Do you understand all the words used in stating the problem? Do you need to ask a question to get the answer? 1. UNDERSTAND THE PROBLEM • First. You have to understand the problem. • What is the unknown? What are the data? What is the condition? • Is it possible to satisfy the condition? Is the condition sufficient to deter- mine the unknown? Or is it insufficient? Or redundant? Or contradictory? • Draw a figure. Introduce suitable notation. • Separate the various parts of the condition. Can you write them down?

Critical thinking is variously defined as:

"the process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and evaluating information to reach an answer or conclusion"[5] "disciplined thinking that is clear, rational, open-minded, and informed by evidence"[5] "reasonable, reflective thinking focused on deciding what to believe or do"[6] "purposeful, self-regulatory judgment which results in interpretation, analysis, evaluation, and inference, as well as explanation of the evidential, conceptual, methodological, criteriological, or contextual considerations upon which that judgment is based"[7] "includes a commitment to using reason in the formulation of our beliefs"[8] in critical social theory, it is the commitment to the social and political practice of participatory democracy; willingness to imagine or to remain open to considering alternative perspectives; willingness to integrate new or revised perspectives into our ways of thinking and acting; and willingness to foster criticality in others.[9] the skill and propensity to engage in an activity with reflective scepticism (McPeck, 1981) disciplined, self-directed thinking which exemplifies the perfection of thinking appropriate to a particular mode or domain of thinking (Paul, 1989, p. 214) thinking about one's thinking in a manner designed to organize and clarify, raise the efficiency of, and recognize errors and biases in one's own thinking. Critical thinking is not 'hard' thinking nor is it directed at solving problems (other than 'improving' one's own thinking). Critical thinking is inward-directed with the intent of maximizing the rationality of the thinker. One does not use critical thinking to solve problems—one uses critical thinking to improve one's process of thinking.[10]

How to Solve It

- First, you have to understand the problem. - After understanding, then make a plan. - Carry out the plan. - Look back on your work. How could it be better?

Problem solving (computer science)

In computer science and in the part of artificial intelligence that deals with algorithms ("algorithmics"), problem solving encompasses a number of techniques known as algorithms, heuristics, root cause analysis, etc. In these disciplines, problem solving is part of a larger process that encompasses problem determination, de-duplication, analysis, diagnosis, repair, etc.

Innovation

Innovation is defined simply as a "new idea, device, or method".[1] However, innovation is often also viewed as the application of better solutions that meet new requirements, unarticulated needs, or existing market needs.[2] This is accomplished through more-effective products, processes, services, technologies, or business models that are readily available to markets, governments and society. The term "innovation" can be defined as something original and more effective and, as a consequence, new, that "breaks into" the market or society.[3] It is related to, but not the same as, invention.[4] While a novel device is often described as an innovation, in economics, management science, and other fields of practice and analysis, innovation is generally considered to be the result of a process that brings together various novel ideas in a way that they affect society. In industrial economics, innovations are created and found empirically from services to meet the growing consumer demand.[5][6]

Intuition

Intuition is the ability to acquire knowledge without proof, evidence, or conscious reasoning, or without understanding how the knowledge was acquired.[1][2] Different writers give the word "intuition" a great variety of different meanings, ranging from direct, mystic insight to unconscious pattern-recognition.[3][4] The word "intuition" is often misused or misunderstood to mean instinct, truth, belief, meaning, and other subjects.[5]

Creative Problem Solving Techniques

Mental state shift: Creativity techniques designed to shift a person's mental state into one that fosters creativity. These techniques are described in creativity techniques. One such popular technique is to take a break and relax or sleep after intensively trying to think of a solution. Problem reframing: Creativity techniques designed for reframing the problem. For example, reconsidering one's goals by asking "What am I really trying to accomplish?" can lead to useful insights. Multiple idea facilitation: Creativity techniques designed to increase the quantity of fresh ideas. This approach is based on the belief that a larger number of ideas increases the chances that one of them has value. Some of these techniques involve randomly selecting an idea (such as choosing a word from a list), thinking about similarities with the undesired situation, and hopefully inspiring a related idea that leads to a solution. Such techniques are described in creativity techniques. Inducing change of perspective: Creative-problem-solving techniques designed to efficiently lead to a fresh perspective that causes a solution to become obvious. This category is especially useful for solving especially challenging problems.[2] Some of these techniques involve identifying independent dimensions that differentiate (or separate) closely associated concepts.[3] Such techniques can overcome the mind's instinctive tendency to use "oversimplified associative thinking" in which two related concepts are so closely associated that their differences, and independence from one another, are overlooked.[4]

Second principle: Devise a plan

Polya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included: Guess and check Make an orderly list Eliminate possibilities Use symmetry Consider special cases Use direct reasoning Solve an equation Look for a pattern Draw a picture Solve a simpler problem Use a model Work backward Use a formula Be creative Use your head/noggin

Fourth principle: Review/extend

Pólya25 mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem. 4. LOOKING BACK • Fourth. Examine the solution obtained. • Can you check the result? Can you check the argument? • Can you derive the solution differently? Can you see it at a glance? • Can you use the result, or the method, for some other problem?

Polya's heuristics

The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:

Core Critical Thinking Skills

The list of core critical thinking skills includes observation, interpretation, analysis, inference, evaluation, explanation, and metacognition. According to Reynolds (2011), an individual or group engaged in a strong way of critical thinking gives due consideration to establish for instance:[11] Evidence through reality Context skills to isolate the problem from context[clarification needed] Relevant criteria for making the judgment well Applicable methods or techniques for forming the judgment Applicable theoretical constructs for understanding the problem and the question at hand In addition to possessing strong critical-thinking skills, one must be disposed to engage problems and decisions using those skills. Critical thinking employs not only logic but broad intellectual criteria such as clarity, credibility, accuracy, precision, relevance, depth, breadth, significance, and fairness.[12]

Problem Solving Strategies

Abstraction: solving the problem in a model of the system before applying it to the real system Analogy: using a solution that solves an analogous problem Brainstorming: (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum solution is found Divide and conquer: breaking down a large, complex problem into smaller, solvable problems Hypothesis testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption Lateral thinking: approaching solutions indirectly and creatively Means-ends analysis: choosing an action at each step to move closer to the goal Method of focal objects: synthesizing seemingly non-matching characteristics of different objects into something new Morphological analysis: assessing the output and interactions of an entire system Proof: try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it Reduction: transforming the problem into another problem for which solutions exist Research: employing existing ideas or adapting existing solutions to similar problems Root cause analysis: identifying the cause of a problem Trial-and-error: testing possible solutions until the right one is found

Analytical Skills

Analytical skill is the ability to visualize, articulate, conceptualize or solve both complex and uncomplicated problems by making decisions that are sensible given the available information. Such skills include demonstration of the ability to apply logical thinking to breaking complex problems into their component parts[clarification needed]. In 1999, Richards J. Heuer Jr., explained that: "Thinking analytically is a skill like carpentry or driving a car. It can be taught, it can be learned, and it can improve with practice. But like many other skills, such as riding a bike, it is not learned by sitting in a classroom and being told how to do it. Analysts learn by doing."[1] To test for analytical skills one might be asked to look for inconsistencies in an advertisement, put a series of events in the proper order, or critically read an essay[citation needed]. Usually standardized tests and interviews include an analytical section that requires the examiner to use their logic to pick apart a problem and come up with a solution. Although there is no question that analytical skills are essential, other skills are equally required[clarification needed]. For instance in systems analysis the systems analyst should focus on four sets of analytical skills: systems thinking, organizational knowledge, problem identification, and problem analyzing and solving.

Association

Association in psychology refers to a connection between conceptual entities or mental states that results from the similarity between those states or their proximity in space or time. The idea stems from Plato and Aristotle, especially with regard to the succession of memories, and it was carried on by philosophers such as John Locke, David Hume, David Hartley, James Mill, John Stuart Mill, and Alexander Bain.[1] It finds its place in modern psychology in such areas as conditioning and in neural network models of memory.[2]

Critical Thinking

Critical thinking, also called critical analysis, is clear, rational thinking involving critique. Its details vary amongst those who define it. According to Barry K. Beyer (1995), critical thinking means making clear, reasoned judgments. During the process of critical thinking, ideas should be reasoned, well thought out, and judged.[1] The National Council for Excellence in Critical Thinking[2] defines critical thinking as the intellectually disciplined process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and/or evaluating information gathered from, or generated by, observation, experience, reflection, reasoning, or communication, as a guide to belief and action.'[3] Critical thinking is variously defined as: "the process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and evaluating information to reach an answer or conclusion"[5] "disciplined thinking that is clear, rational, open-minded, and informed by evidence"[5] "reasonable, reflective thinking focused on deciding what to believe or do"[6] "purposeful, self-regulatory judgment which results in interpretation, analysis, evaluation, and inference, as well as explanation of the evidential, conceptual, methodological, criteriological, or contextual considerations upon which that judgment is based"[7] "includes a commitment to using reason in the formulation of our beliefs"[8] in critical social theory, it is the commitment to the social and political practice of participatory democracy; willingness to imagine or to remain open to considering alternative perspectives; willingness to integrate new or revised perspectives into our ways of thinking and acting; and willingness to foster criticality in others.[9] the skill and propensity to engage in an activity with reflective scepticism (McPeck, 1981) disciplined, self-directed thinking which exemplifies the perfection of thinking appropriate to a particular mode or domain of thinking (Paul, 1989, p. 214) thinking about one's thinking in a manner designed to organize and clarify, raise the efficiency of, and recognize errors and biases in one's own thinking. Critical thinking is not 'hard' thinking nor is it directed at solving problems (other than 'improving' one's own thinking). Critical thinking is inward-directed with the intent of maximizing the rationality of the thinker. One does not use critical thinking to solve problems—one uses critical thinking to improve one's process of thinking.[10]

Invention

If an innovation is unique, original and novel, it is considered an invention. It must be original - not known to people who are knowledgeable in the field of the solution, and novel - not an obvious solution, easily conceived by people with knowledge in the field of that solution - when shown the problem. Not all inventions are created through creative problem-solving. Inventions may be 'discovered' or 'contemplated', many times without a 'problem to solve', or solving a problem that they originally were not intended for. But many inventions in fact are the outcome of creative problem-solving. An invention can become intellectual property if the inventor files and receives a patent, which is a legal document, where the invention is clearly defined, and which shows proof of its uniqueness, originality and novelty.