The Essentials of Problem Solving
An introduction to the psychology of problem solving. Featured problems: the towers of Hanoi, the Chinese ring puzzle, the Wason 4-card selection task, the candle problem, Roman matchstick problems, and toothpick shape problems. 00:00 A quick note 00:47 The problem state space and the towers of Hanoi 4:45 Problems of representation and the Chinese ring puzzle 6:42 Context and variations of the Wason 4-card selection task 9:42 Introduction to insight problems: the candle problem 11:05 Differences between insight and incremental problems 12:15 Barriers to insight: Roman matchstick problems 17:30 Insight problems: too big of a distinction? 19:08 Well-structured and ill-structured problems 21:11 Representation and argument 23:34 Becoming a better problem solver: toothpick problems 26:45 Domain-specific knowledge and strategy change 30:55 What transfers across problem-solving domains? For more videos on learning, check out my online learning community: https://www.benjaminkeep.com/community Newsletter: www.benjaminkeep.com ACKNOWLEDGEMENTS The towers of Hanoi footage comes from Math Playground's version. Chinese ring puzzle footage comes from Youtuber @Jojikiba, here https://www.youtube.com/watch?v=nydIl2VO12I The candle problem visuals: https://www.youtube.com/watch?v=PEvThYxtnFQ. If know who made it, please let me know. The link above seems to be the oldest version. Paul Zeitz photo: https://www.imo-official.org/advisory.aspx?year=2019 Math and STEP problems: https://www.imo-official.org/problems/IMO2022SL.pdf and https://www.usmle.org/sites/default/files/2021-10/Step_1_Sample_Items.pdf, respectively. Physics problems: Badeau, R., White, D. R., Ibrahim, B., Ding, L., & Heckler, A. F. (2017). What works with worked examples: Extending self-explanation and analogical comparison to synthesis problems. Physical Review Physics Education Research, 13(2), 020112. https://doi.org/10.1103/PhysRevPhysEducRes.13.020112 Visit www.eternagame.org for more info on Eterna. The Chess explanation clip: @BetterChessTraining. The Starcraft 2 clip: an @ESChamp video, here: https://www.youtube.com/watch?v=4okBbHaCQ98 Go examples come from Sensei’s library: https://senseis.xmp.net/?CornersThenSidesThenCenter Starcraft 2 map: https://news.blizzard.com/en-us/starcraft2/23223597/new-ladder-maps-for-2019-season-4 Parity example: https://artofproblemsolving.com/wiki/index.php?title=Parity The free body diagram is from a Math and Science: https://www.youtube.com/watch?v=mI11vHsRnC8 Map of Easter Island: https://en.wikipedia.org/wiki/File:Easter_Island_map-en.svg The cashflow statement: https://en.wikipedia.org/wiki/Cash_flow_statement#cite_note-21, but is originally from Epstein, Barry J.; Eva K. Jermakowicz (2007). Interpretation and Application of International Financial Reporting Standards. John Wiley & Sons. pp. 91–97. ISBN 978-0-471-79823-1. REFERENCES Simon, H. A., & Newell, A. (1971). Human problem solving: The state of the theory in 1970. American psychologist, 26(2), 145. The tower of Hanoi state space was adapted from Zhang, J., & Norman, D. A. (1994). Representations in distributed cognitive tasks. Cognitive science, 18(1), 87-122. The Chinese Ring puzzle state space (for 5 rings) is adapted from Kotovsky, K. & Simon, H. A. (1990). What Makes Some Problems Really hard: Explorations in the Problem Space of Difficulty. Cognitive Psychology, 22(2), 143–183. The “underage drinking” example of the Wason 4-card task comes from this classic: Griggs, R. A., & Cox, J. R. (1982). The elusive thematic‐materials effect in Wason's selection task. British journal of psychology, 73(3), 407-420. A short meta-analysis on the Wason 4-card task: Ragni, M., Kola, I., & Johnson-Laird, P. (2017). The Wason Selection task: A Meta-Analysis. In CogSci. https://www.researchgate.net/profile/Ilir-Kola/publication/322682384_The_Wason_Selection_Task_A_Meta-Analysis/links/5a688e6f4585156abdffd541/The-Wason-Selection-Task-A-Meta-Analysis.pdf On the insight experience: Webb, M. E., Little, D. R., & Cropper, S. J. (2016). Insight Is Not in the Problem: Investigating Insight in Problem Solving across Task Types. Frontiers in Psychology, 7. https://www.frontiersin.org/articles/10.3389/fpsyg.2016.01424 The Matchstick Roman numeral problems and the discussion of constraints and chunks come from: Öllinger, M., Jones, G., & Knoblich, G. (2008). Investigating the Effect of Mental Set on Insight Problem Solving. Experimental Psychology, 55(4), 269–282. https://doi.org/10.1027/1618-3169.55.4.269 On strategic change, see: Siegler, R. S. (2002). Microgenetic studies of self-explanation. Microdevelopment: Transition processes in development and learning, 31, 58. On the importance of representations in ill-structured domains, see: Chandrasekharan, S., & Nersessian, N. J. (2011). Building cognition: the construction of external representations for discovery. In Proceedings of the Annual Meeting of the Cognitive Science Society (Vol. 33, No. 33).
The Essentials of Problem Solving
Introduction to Problem Solving
In this section, the speaker introduces the concept of problem solving and its importance in becoming a better problem solver. They also mention that they will discuss classic problems used by psychologists to study problem solving.
Understanding Problem Solving
- Learning more about problem solving can make you a better problem solver.
- Herbert Simon and Alfred Newell's formulation of problem solving is well-studied.
- The Towers of Hanoi puzzle is used as an example to explain different aspects of problem solving.
- The puzzle has a starting state, operations that change the state, and a goal state.
- The concept of a problem state space is introduced, which describes all the states and operations of a problem.
Complexity of Problem Solving
- The complexity of a problem depends on the number of states, paths, and possible operations in its state space.
- Some problems may be unsolvable or have multiple solutions.
- The idea of state space applies not only to puzzles but also to two-player games like chess.
Chinese Ring Puzzle
- The Chinese ring puzzle is introduced as an example where understanding the current state and effective movement between states is crucial for solving the puzzle.
- Unlike chess, where you can see all the pieces' positions, the ring puzzle requires clear understanding of the current state to solve it effectively.
Interpreting Problem Spaces
- Problem solvers' interpretation of a problem space affects how they navigate through it.
- A card-turning problem and a police officer's job are mentioned as examples where interpreting the problem space correctly leads to effective solutions.
Conclusion
In this final section, key insights about problem solving are summarized.
Key Insights
- Learning about different aspects of problem solving can improve one's ability to solve problems.
- Problem solving involves understanding the problem space, including the starting state, operations, and goal state.
- The complexity of a problem is determined by the number of states, paths, and possible operations in its state space.
- Effective navigation through the problem space depends on how one interprets and understands it.
The transcript provided does not have specific timestamps for each bullet point. I have associated the bullet points with approximate timestamps based on their position in the transcript.
The Wason Card Selection Task
This section discusses the Wason Card Selection Task, a well-studied puzzle in psychology that explores how and why people choose the wrong cards. It highlights the common mistake of choosing the vowel card and even card, despite them not providing relevant information to evaluate a given statement.
Understanding the Wason Card Selection Task
- The Wason Card Selection Task is a puzzle that involves determining which cards need to be turned over to determine if people are violating a rule.
- Most people tend to choose the vowel card and even card, assuming they provide relevant information. However, this is incorrect as the statement being evaluated does not mention anything about consonants.
- The correct card to pick is actually the seven because if it has a vowel on the back, it violates the rule. If it has an odd number on the back, it also violates the statement.
- People often have an easier time understanding this problem compared to other similar tasks due to real-world context and familiarity with rules like drinking age restrictions.
Factors Affecting Problem Solving
This section explores three factors that influence problem-solving: 1) The complexity of the state space, 2) Interpretation of the problem, and 3) Contextual influences.
Factors Affecting Problem Solving
- Problem-solving is influenced by three main factors:
- The complexity of the state space itself and how complex it is.
- Our interpretation of the problem, including understanding our current state and how to transition to a different state.
- The context in which we approach and think about the problem.
Insight Problems vs Incremental Problems
This section distinguishes between insight problems and incremental problems in problem-solving. Insight problems involve sudden realizations or "aha" moments, while incremental problems require building up the solution incrementally.
Insight Problems vs Incremental Problems
- Insight problems are characterized by sudden realizations or "aha" moments that lead to solving the problem.
- These problems are different from incremental problems, where solutions are built up gradually.
- In insight problems, people often feel stuck and repeatedly explore possibilities that don't work until they suddenly arrive at a solution.
- Incremental problems allow individuals to gauge their progress towards a solution, while insight problems lack this sense of progress.
Barriers to Insight: Chunks and Constraints
This section discusses two barriers to insight: 1) Chunks, where we struggle to move between different levels of perception, and 2) Constraints, where our initial representation of the problem contains implicit assumptions that limit our search space.
Chunks and Constraints in Problem Solving
- Chunks refer to perceiving certain elements as wholes or units rather than breaking them down into their component parts.
- Moving between different levels of chunks can be challenging in problem-solving.
- Constraints arise when our initial representation of the problem includes implicit assumptions that restrict our search for solutions.
- Overcoming constraints involves letting go of these assumptions and exploring alternative possibilities.
Roman Numeral Match Stick Problems
This section introduces Roman numeral match stick problems as an example illustrating barriers to insight. It demonstrates how decomposing chunks and overcoming constraints are essential for solving such problems.
Roman Numeral Match Stick Problems
- Roman numeral match stick problems involve manipulating match sticks arranged as Roman numerals to create true equations.
- Decomposing chunks is crucial in solving these problems. For example:
- Moving a single matchstick can transform IV (four) into VI (six).
- Rearranging matchsticks forming an X can create a V (five).
- Constraints play a role in these problems, as initial representations may limit the search for solutions.
- Overcoming constraints involves challenging assumptions and exploring alternative arrangements of matchsticks.
The transcript does not provide timestamps beyond this point.
New Section
This section discusses the concept of mental set and how problem-solving strategies can be biased based on previous successful approaches.
Mental Set and Problem Solving Strategies
- Mental set refers to the tendency to stick to a particular problem-solving strategy that has been successful in the past.
- Solving a series of problems successfully with the same approach can impair the ability to see alternative strategies when faced with a new problem.
- Certain chunks and constraints in problem-solving can be easier or harder for individuals.
- Manipulating problem representations during problem solving can create biased perspectives.
New Section
This section explores how mental set affects problem solving and insight experiences.
Impact of Mental Set on Problem Solving
- When a strategy continues not to work, individuals may reach an impasse and expand their thinking to consider other potential strategies or solutions.
- Incremental and insight problems are traditionally considered as two different types, but whether someone has an insight experience depends on more than just the nature of the problem.
- People may report having insight experiences even when solving what would traditionally be called incremental problems.
New Section
This section discusses how incremental and insight elements are present in math and science problems.
Incorporating Incremental and Insight Elements
- Many math and science problems incorporate both incremental (analytic) elements and insight (conceptual) elements together.
- Some math problems involve "Crux moves," which require insights to solve, but finding these moves does not eliminate the need for hard work in finding the solution.
- Realistic problems often have elements of both incremental and insight processes.
New Section
This section highlights that most realistic problems are ill structured, unlike well structured problems studied by psychologists.
Well Structured vs. Ill Structured Problems
- Well structured problems have well-defined goal states, fixed operations, and a clear state space.
- Ill structured problems, such as addressing climate change or providing public transportation systems, lack precise goal states and involve complex interactions with poorly defined elements.
- Lessons from well structured problems still apply to ill structured problems, including the need to evaluate alternatives and make tradeoffs.
New Section
This section emphasizes the importance of problem representation in ill structured problems.
Problem Representation in Ill Structured Problems
- Understanding the nature of ill structured problems plays a significant role in finding solutions.
- Unlike insight experiences where the right representation leads to an obvious solution, it is not immediately clear if a given representation will be helpful in ill structured problems.
- Different representations or combinations of representations can simplify or complicate problem-solving processes.
New Section
This section discusses prioritizing outcomes and dealing with changing circumstances in solving ill structured problems.
Prioritizing Outcomes and Dealing with Change
- Ill structured problems involve prioritizing outcomes since there is no universally agreed-upon goal state.
- Solutions often require tradeoffs between different outcomes.
- Real-world problems are subject to changing circumstances and evolving standards, which may render previously considered good solutions ineffective or problematic in the long term.
New Section
This section highlights that well structured problems serve as models for understanding problem-solving dynamics but may not fully represent real-world complexities.
Well Structured Problems as Models
- Well structured problems provide interesting models for studying problem-solving dynamics.
- Simplifying these models helps isolate specific cognitive processes but comes with tradeoffs in representing real-world complexities.
New Section
In this section, the speaker discusses problem-solving strategies using a toothpick problem as an example.
Becoming a Better Problem Solver
- The speaker introduces a toothpick problem where the goal is to remove five toothpicks and leave three squares.
- Initially, trial and error is used to explore different possibilities.
- The speaker explains the concept of domain general strategy, which involves trying out different approaches without specific guidelines.
- Two attempts are made based on the prompt of removing five toothpicks, but they do not yield the desired solution.
- A more systematic approach is suggested by removing one toothpick at a time to observe the resulting shapes.
- Different types of pieces (corners, edges, centers) are identified and their impact on creating squares is analyzed.
- Domain-specific knowledge is gained through this process, allowing for more efficient problem-solving in similar situations.
New Section
This section explores how problem-solving strategies can be applied to different domains using RNA molecule design as an example.
Applying Problem-Solving Strategies to RNA Design
- The speaker introduces the problem of designing RNA molecules with specific structures based on sequences of bases.
- It is noted that the knowledge gained from solving the toothpick problems does not directly apply to understanding RNA design.
- Different problem domains require specific expertise and understanding.
- Developing expertise in problem-solving usually involves becoming proficient in solving a particular class of problems within a specific domain.
- Expertise in one area may not necessarily transfer well to another area due to differences in problem characteristics and requirements.
Conclusion
Problem-solving strategies can be developed through practice and experience. However, it's important to recognize that expertise in one domain may not directly translate into proficiency in another domain. By gaining domain-specific knowledge and understanding patterns within a particular class of problems, individuals can become more efficient problem solvers within that domain.
The Importance of Domain General Techniques
This section discusses the importance of domain general techniques in problem-solving and their role in understanding different domains effectively.
Transferability of Problem-Solving Skills
- Transfer is the idea that we can apply something learned in one situation to a different situation.
- Certain large ideas, such as the concept of terrain or symmetry, can transfer between problem domains.
- Strategic problem-solving behaviors like organizing knowledge, stating assumptions explicitly, and estimating solution costs also tend to transfer across domains.
- Personality traits like stopping to think before answering and perseverance are beneficial for problem solving in various domains.
Learning from Different Domains
- Understanding specific features of a domain drives high levels of performance.
- While certain ideas and strategies may be useful across many domains, their application depends on the specific characteristics of each domain.
- Experience using different representational tools helps organize knowledge effectively in diverse domains.
The Problem of Getting Better at Problem Solving
- To solve hard problems, learning while solving is crucial. Simply performing exercises without learning does not lead to problem-solving expertise.
- Good problem solvers distinguish themselves by bringing conceptual tools, strategic tools, and a willingness to learn from problems they encounter.
Conclusion
The video concludes with an invitation for questions, comments, and thoughts from viewers.
Feel free to leave any questions or comments in the comment section below.