6: Problem Solving
- Page ID
- 54097
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 6.1: Types of Problems
- The text discusses the distinction between well-defined and ill-defined problems. Well-defined problems have clear initial states, finite rules, and specific goals, making them suitable for algorithmic solutions, exemplified by tasks like playing chess. In contrast, ill-defined problems, such as creative tasks, lack clear states or goals, making formalization difficult. However, ill-defined problems often contain subproblems that are well-defined, requiring both structured and creative thinking.
- 6.2: Problem Solving Strategies
- The text discusses various problem-solving strategies essential for tackling issues, whether simple or complex. Key strategies include trial and error, algorithm, and heuristic methods. Trial and error involves trying multiple solutions until success, while algorithms provide step-by-step problem-solving formulas. Heuristics offer a general framework, employing shortcuts or rules of thumb to simplify decision-making.
- 6.3: Means–Ends Analysis
- Means-End Analysis is a problem-solving technique that involves creating subgoals to bridge the gap between an initial state and a goal state, often utilizing recursion. An example is the "Towers of Hanoi," where discs are moved from the start peg to the end peg following specific rules. This method can be easily implemented in algorithms and is applicable in everyday situations like planning train connections by solving subproblems sequentially to reach the desired endpoint.
- 6.4: Reasoning by Analogy
- The text discusses analogical problem solving, a technique for finding solutions to problems by using analogous solutions from other problems. It explains how Gick and Holyoak investigated this method using Duncker's radiation problem and the General's story. Their study highlighted the three steps crucial for analogical problem solving: noticing connections, mapping problem elements, and applying the mapping to solve the target problem.
- 6.5: Transformation Problems
- Transformation problems in cognitive psychology involve significant changes in an individual's thought or behavior patterns, requiring a sequence of transformations to achieve specific goals. This concept can be exemplified by the Wallas Stage Model of the Creative Process, which outlines stages such as preparation, internalization/incubation, illumination, and verification/elaboration in creative problem-solving.
- 6.6: Incubation
- Incubation involves taking a break from actively solving a problem to let unconscious processes work, potentially improving problem-solving outcomes. This can include activities like sleeping or taking on different tasks. Research indicates that low-level cognitive tasks enhance problem-solving more effectively than complex tasks. In educational settings, breaks have been shown to boost children's creativity and problem-solving capabilities.
- 6.7: Problem Solving Experts
- Experts are individuals who achieve mastery in a specific field through significant dedication and effort. They excel in solving field-related problems due to possessing extensive knowledge, a distinct organization of this knowledge, and a strategic approach to problem analysis. Unlike novices, experts recognize patterns and utilize deep structure over surface similarities, as demonstrated in studies by Chase and Simon on chess players and Chi on physics problems.
- 6.8: Blocks to Problem Solving
- The text discusses how fixation, particularly functional fixedness, impedes problem solving by limiting individuals to traditional uses of objects. It illustrates this with examples like the candle and two-string problems where habitual uses hinder novel solutions. Mental fixedness is linked to incorrect problem-solving approaches based on past experiences, leading to challenges like incorrect answer persistence.