THINKING 1

Problem Solving and Functional Fixedness

• Mental Set and Priming in Cognition:

  • Cognitive processing can become rapidly biased toward specific patterns through exposure to immediate phonological or conceptual primes.

  • Phonological Priming Demonstration:

    • When presented with the spelling and pronunciation pattern of words like "oak" and asked for a synonym for people or family that fits this pattern, individuals retrieve "folk."

    • When asked to identify the American president in office during the Mexican-American War, individuals retrieve "James K. Polk."

    • When subsequently asked to identify the word that means "egg white," individuals systematically respond with "yolk," despite knowing factually that yolk is the yellow center of an egg and albumen is the scientific term for egg white.

    • This error demonstrates how rapidly an activated cognitive track (a mental set) constrains search retrieval, overriding basic factual knowledge.

• Definition of Functional Fixedness:

  • Functional fixedness is a cognitive bias that limits a person to using an object only in the way it is traditionally or customarily used.

  • Overcoming functional fixedness requires cognitive restructuring, enabling an individual to see beyond an object's primary identity and leverage its physical properties for alternative tasks (e.g., using a freshly emptied wine bottle as an impromptu rolling pin to flatten pizza dough when a standard rolling pin is unavailable).

• The Two-String Problem:

  • Experimental Setup:

    • A participant enters a laboratory room where two strings hang down from the ceiling.

    • Available items inside the room include a chair, a pair of pliers, a jar of tacks, and sheets of paper.

    • Task: Tie the two hanging strings together.

    • The Dilemma: The strings are spaced far enough apart that holding onto one string prevents reaching the second string.

  • Typical Ineffective Approaches:

    • Standing on the chair: This elevates the participant vertically but fails to reduce the horizontal distance between the strings.

    • Attempting to elongate the string using external objects (e.g., tearing paper to extend the reach) or bringing outside materials into the room (which violates test constraints).

  • The Standard Target Solution:

    • Overcoming functional fixedness regarding the pliers.

    • Instead of using pliers for gripping or cutting, the subject ties the pliers to the end of one string, creating a pendulum.

    • The participant swings the weighted string, walks over to hold the second string, waits for the swinging string to arc toward them, catches it, and ties them together.

  • Performance Data:

    • Given a period of $10$ minutes without hints, only $39\%$ of naive participants successfully solve the problem.

• The Candle Problem and Object Functionality:

  • Experimental Setup:

    • Subjects are given a candle, a box of matches, and a box of thumbtacks.

    • Task: Affix the candle to a closed wooden door such that wax will not drip onto the surface below.

  • Typical Failed Attempts:

    • Melting the side or bottom of the candle wax with a match and attempting to adhere the candle directly to the door.

    • This fails because candle wax lacks the structural adhesion necessary to hold the weight of the candle against a vertical surface.

  • Target Solution:

    • Empty the thumbtacks out of their container box.

    • Tack the cardboard box directly to the door using the thumbtacks.

    • Place the candle upright inside the tacked box.

  • Performance Data:

    • Fewer than $1$ in $4$ participants ($< 25\%$) spontaneously produce this solution when the tacks are presented inside the box.

  • The "Unemployed Box" Effect:

    • When experimenters alter the condition by dumping the tacks out prior to the test and presenting an empty box next to the loose tacks, problem difficulty plummets.

    • Nearly all participants successfully tack the empty box to the door.

    • Theoretical Insight: When an object is actively serving a designated function (e.g., a box functioning as a tack container), its identity as a container conceals its potential identity as a structural shelf. Freeing an object from an active function makes its raw affordances visible.

• Wittgenstein's Broom Metaphor:

  • Philosopher Ludwig Wittgenstein posed an epistemological question regarding a broom composed of two elements: a handle and bristles.

  • The Question: Which part represents the essential or most important component of the broom?

  • Context-Dependent Utility: While intuition designates the bristles as the essence of "broomness," if the operational goal shifts to breaking a glass window to gain entry to a house, the structural wooden handle becomes infinitely more functional than the bristles.

  • The utility and importance of an object's component parts are fundamentally governed by the situational goal rather than by abstract essentialism.

• The Nine-Dot Problem and Self-Imposed Boundaries:

  • Setup and Rules:

    • A grid of nine dots ($3 \times 3$ array) is presented on paper.

    • The task requires connecting all nine dots using exactly $4$ straight, continuous lines without lifting the pen from the paper.

  • The Core Cognitive Obstacle:

    • Problem solvers implicitly generate a non-existent rule: that lines must remain strictly within the square perimeter defined by the nine dots.

    • This problem cannot be solved if the lines remain bounded within the perimeter.

  • Progressive Hinting Data:

    • In experimental evaluations, the baseline control group (individuals who fail to solve it spontaneously within the initial allotted time) has a success rate of $0\%$.

    • When given the explicit hint that they are permitted to draw outside the boundary of the square, roughly $25\%$ of failed participants solve it.

    • When told they can draw outside the square and the experimenter draws the first line for them, slightly over $50\%$ solve it.

    • When told they can draw outside the square and the experimenter draws the first two lines, virtually $100\%$ solve the puzzle.

    • Knowing abstractly that boundaries may be broken is often insufficient; procedural mental sets require concrete operational demonstrations to be fully dismantled.

Analogical Problem Solving and Structural Transfer

• The Radiation (Tumor) Problem:

  • Scenario:

    • A patient has an inoperable, lethal stomach tumor.

    • A specific type of ray can destroy the tumor at a high intensity.

    • High-intensity rays simultaneously destroy intervening healthy tissue, which will kill the patient.

    • Low-intensity rays pass harmlessly through healthy tissue but have zero destructive effect on the tumor.

    • Task: Devise a method to destroy the tumor with the rays without damaging the surrounding healthy tissue.

  • Target Solution:

    • Direct multiple low-intensity rays from different angles toward the body such that they converge at the exact anatomical site of the tumor.

    • The cumulative intensity at the point of convergence reaches lethal therapeutic thresholds for the tumor while individual ray paths through healthy tissue remain harmless.

  • Baseline Performance:

    • Only approximately $10\%$ ($1$ in $10$) of unguided participants generate this convergence solution spontaneously.

• The Fortress Problem (Analogical Counterpart):

  • Scenario:

    • A military general seeks to capture a centrally located fortress.

    • All radial approach roads are heavily mined to detonate if an entire army marches over them together.

    • Small detachments of troops can traverse the roads without triggering the mines.

  • Deep Structural Solution:

    • Divide the large army into small units, march them simultaneously down the multiple radial roads from different directions, and converge on the fortress at the same moment.

• Deep Structure versus Surface Structure:

  • Surface Structure:

    • The superficial narrative domain and contextual elements (e.g., oncology, ionizing radiation, stomach tissue vs. warfare, generals, fortresses, landmines).

  • Deep Structure:

    • The underlying logical architecture of the problem: a central target requiring concentrated energy/force surrounded by paths that fail or suffer catastrophic damage if traversed by high-magnitude force all at once.

  • Failure of Spontaneous Analogical Transfer:

    • When participants solve or read the solution to the Tumor Problem and immediately proceed to the Fortress Problem, only approximately $33\%$ ($1$ in $3$) spontaneously transfer the underlying logical structure to solve the second problem.

    • When participants are explicitly told beforehand that the second problem shares a deep logical structure with the first, performance jumps to virtually $100\%$.

  • Implications for Real-World Reasoning:

    • Human cognition relies heavily on superficial perceptual cues to determine similarity.

    • In political and military rhetoric, complex geopolitical situations are frequently equated based on superficial traits (e.g., declaring any military intervention to be "just like Vietnam" simply because both involve warfare), while the structural, operational, and historical realities share zero deep equivalence.

Judgmental Heuristics: Kahneman and Tversky

• Heuristics versus Algorithms:

  • Algorithm:

    • A formal, systematic, step-by-step mathematical or logical procedure that is guaranteed to yield a correct answer if executed properly (e.g., calculating circumferences using $C = 2\pi r$).

    • Algorithms are cognitively demanding, slow, and computationally expensive.

  • Heuristic:

    • A mental shortcut, rule of thumb, or intuitive visual/cognitive strategy that allows rapid problem solving and probabilistic estimation.

    • Heuristics are evolutionarily beneficial and usually functional, but systematically produce predictable cognitive biases and errors under specific conditions.

• The Representativeness Heuristic:

  • Definition: The tendency to judge the probability or frequency of an event by the degree to which it resembles its parent population or reflects the salient features of the process that generates it.

  • Coin Toss Probability Example:

    • Consider a fair coin flipped $6$ times. Three potential outcomes:

      1. $\text{H - H - H - H - H - H}$

      2. $\text{H - H - H - T - T - T}$

      3. $\text{T - H - H - T - H - T}$

    • Theoretical Reality: In a fair coin toss where each flip has $P = 0.5$, every specific, ordered sequence of $6$ flips has an identical mathematical probability:             $P = \left(\frac{1}{2}\right)^6 = \frac{1}{64}$

    • Cognitive Bias: People overwhelming select the third sequence as significantly more probable because its disordered appearance is representative of their mental model of a "random process." The first sequence appears systematically engineered and non-random.

  • Male Height Sample Example:

    • At a university where the average male height is $175\,\text{cm}$, three men are randomly sampled.

    • Outcome A: Three men with varied heights containing a noticeable outlier.

    • Outcome B: Three men whose heights are all extremely close to $175\,\text{cm}$.

    • Cognitive Bias: People intuitively select Outcome A as more probable because a random draw is expected to look variable.

    • Statistical Reality: Assuming a normal (Gaussian) distribution, the probability density is maximal at the mean; drawing observations clustered tightly around $175\,\text{cm}$ is mathematically more probable than drawing points with extreme variance.

  • The Conjunction Fallacy (The Linda Problem):

    • Profile: Linda is $31$ years old, single, outspoken, highly intelligent, majored in philosophy, was deeply concerned with issues of discrimination and social justice as a student, and participated in antinuclear demonstrations.

    • Test Options Evaluated:

      • Option D: Linda is a bank teller.

      • Option F: Linda is a bank teller and is active in the feminist movement.

    • Experimental Finding: A vast majority of subjects rank Option F as more probable than Option D.

    • Mathematical Violation: The probability of a conjunction of two independent or dependent events ($A \cap B$) cannot exceed the probability of one of its constituent events ($A$):             $P(D \cap F) \le P(D)$

    • Because active feminism is highly representative of the character sketch of Linda, the cognitive mind prioritizes semantic fit over basic set theory, committing the conjunction fallacy.

• The Availability Heuristic:

  • Definition: The cognitive tendency to estimate the frequency, likelihood, or probability of an event based on the ease with which relevant instances come to mind.

  • Letter Position Frequency:

    • Question: In the English language, does the letter "K" appear more frequently in the first position of a word or in the third position?

    • Empirical Fact: The letter "K" appears approximately twice as often in the third position (e.g., "bike", "make", "rake") as it does in the first position.

    • Human Estimate: When surveyed, participants estimate the exact opposite ratio, asserting that words beginning with "K" are twice as common.

    • Cognitive Architecture Explanation: Human mental lexicons are cataloged alphabetically by initial letters, like a physical dictionary. Searching memory for words that begin with "K" yields rapid, fluid exemplars, whereas searching for words with "K" in the third position is computationally inefficient. Because initial-"K" words are more available to conscious retrieval, their empirical frequency is grossly overestimated.

• Base Rate Neglect in Risk Communication:

  • Medical professionals historically struggle with Bayesian probability and risk estimation, frequently miscommunicating diagnostic test outcomes to patients.

  • The Diagnostic Paradox (Syndrome XYZ):

    • Condition Prevalence (Base Rate): $1$ in $1,000$ individuals ($0.1\%$ or $P(\text{Disease}) = 0.001$).

    • Test Sensitivity (True Positive Rate): $100\%$ (the test detects every individual who has the disease).

    • Test False Positive Rate: $5\%$ ($P(\text{Positive} \mid \text{No Disease}) = 0.05$).

    • Question: If a randomly selected individual tests positive, what is the actual probability that this person possesses Syndrome XYZ?

    • Intuitive Error: Untrained individuals and clinicians routinely estimate the probability at $95\%$, conflating the test's specificity with the posterior probability of disease.

    • Mathematical Breakdown: In a random cohort of $1,000$ individuals:

      • $1$ individual has the condition and tests positive ($1$ true positive).

      • $999$ individuals do not have the condition.

      • Of the $999$ healthy individuals, $5\%$ receive false positive results ($\approx 50$ false positives).

      • Total positive tests: $1 + 50 = 51$.

      • True probability of having the disease given a positive test:                 $P(\text{Disease} \mid \text{Positive}) = \frac{1}{51} \approx 2\%$

    • Failure to incorporate base rates leads to profound overestimations of medical risk.

• Anchoring and Adjustment:

  • Definition: A cognitive heuristic where individuals rely excessively on an initial piece of information encountered (the "anchor") to make subsequent numerical estimates, adjusting insufficiently from that anchor.

  • Factorial Estimation Experiment ($8!$):

    • Stanford students were given $5$ seconds to estimate the product of eight integers presented in either ascending or descending order.

    • Ascending Condition: $1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8$

      • Average estimate: slightly over $500$.

    • Descending Condition: $8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

      • Average estimate: slightly over $2,000$.

    • Actual Mathematical Value:             $8! = 40,320$

    • Because subjects only have time to calculate the first few terms, those starting with $8 \times 7 \times 6$ immediately anchor on a high numerical magnitude, leading to significantly higher overall estimates than those starting with $1 \times 2 \times 3$.

  • Arbitrary Real-World Anchoring Studies:

    • Restaurant Pricing Study: Subjects viewed identical photographs of a restaurant facade, altered only by a name in Photoshop: "Cafe 17" versus "Cafe 71." Subjects given "Cafe 71" estimated significantly higher prices for dinner entrees than subjects exposed to "Cafe 17."

    • United Nations Wheel of Fortune Experiment: Participants observed a wheel of fortune marked with numbers from $1$ to $100$ spun to land on either $10$ or $65$. When asked to estimate the percentage of African nations in the United Nations:

      • Participants exposed to the anchor of $65$ gave an average estimate of $45\%$.

      • Participants exposed to the anchor of $10$ gave an average estimate of $24\%$.

      • Even when participants recognize that the anchor is generated entirely at random, it alters numerical estimation.

    • Mississippi River Length: When subjects are asked whether the Mississippi River is longer or shorter than $500\,\text{miles}$, their absolute numerical estimates average close to $1,000\,\text{miles}$. When first asked whether it is longer or shorter than $5,000\,\text{miles}$, their estimates are wildly higher.

Framing Effects and Decision-Making Biases

• Framing Effects:

  • Definition: Cognitive biases where people decide on options based on whether the options are presented with positive or negative connotations (e.g., as a loss or as a gain).

• The Asian/South American Disease Problem:

  • Context: An unusual disease outbreak is expected to kill $600$ people. Two alternative intervention programs are proposed.

  • Gain Framing Condition (Lives Saved):

    • Program A: Exactly $200$ people will be saved.

    • Program B: A $\frac{1}{3}$ probability that all $600$ people will be saved, and a $\frac{2}{3}$ probability that no people will be saved.

    • Result: $72\%$ of respondents choose Program A (risk-averse behavior).

  • Loss Framing Condition (Lives Lost):

    • Program A: Exactly $400$ people will die.

    • Program B: A $\frac{1}{3}$ probability that nobody will die, and a $\frac{2}{3}$ probability that all $600$ people will die.

    • Result: A substantial majority of respondents choose Program B (risk-seeking behavior).

  • Underlying Equivalence: The mathematical expected values ($EV$) of all choices are identical:         $EV(\text{Program A}) = 200\,\text{survivors}$         $EV(\text{Program B}) = \left(\frac{1}{3} \times 600\right) + \left(\frac{2}{3} \times 0\right) = 200\,\text{survivors}$         Simply shifting the semantic frame from "lives saved" to "lives lost" completely reverses decision-making preferences.

• Asymmetric Risk Orientations for Gains and Losses:

  • Monetary Gain Dilemma:

    • Option 1: Receive a guaranteed sum of $75$.

    • Option 2: Roll a die with a $75\%$ probability of winning $100$ and a $25\%$ probability of winning $0$.

    • Finding: Approximately $84\%$ of people select the certain $75$, demonstrating strong risk aversion in the context of positive gains.

  • Monetary Loss Dilemma:

    • Option 1: Pay a guaranteed penalty of $75$.

    • Option 2: Roll a die with a $75\%$ probability of losing $100$ and a $25\%$ probability of losing $0$.

    • Finding: The majority select the risky gamble (Option 2), preferring the probability of avoiding loss over a guaranteed loss.

  • Core Principle: Human psychology is systematically risk-averse when evaluating gains, but risk-seeking when evaluating losses.

• Perspective-Biased Regulatory Framing:

  • Scenario 1: United States Department of Transportation data indicates a particular German vehicle is eight times more likely to kill occupants of another car in a collision than a standard family vehicle. American subjects overwhelmingly support a federal ban on the sale and use of the German car within the United States.

  • Scenario 2: The identical engineering data is presented regarding an American vehicle operating within Germany. When American subjects are asked whether German authorities should ban the American car from German roads, their responses reverse, opposing the ban.

  • The structural risk is identical ($8 \times$ excess mortality), but decisions are swayed by nationalistic identity and perspective framing rather than objective safety metrics.

• Formal Logic versus Intuition: Hempel's Raven Paradox:

  • Philosopher Carl Gustav Hempel (1940s) demonstrated the profound mismatch between formal logic and human intuition using contrapositive logical statements.

  • Hypothesis: "All ravens are black."

  • Contrapositive Equivalent: "All non-black things are non-ravens."

  • Logical Rule: In formal logic, any evidence confirming a statement equally confirms its logically equivalent contrapositive.

  • Intuitive Failure: Observing a black raven intuitively feels like direct evidentiary support for "All ravens are black." However, observing a green tennis ball (a non-black object that is a non-raven) provides mathematically and logically identical support for the hypothesis. Human intuition rejects the relevance of the green tennis ball entirely, illustrating that the brain is not hardwired to function as a formal logical processing board without explicit training.

Neuropsychology of Thinking: Frontal Lobe and Prefrontal Function

• The Prefrontal Cortex and Executive Control:

  • Higher-order problem solving, abstract reasoning, strategic estimation, and behavioral inhibition are localized largely within the frontal lobes, particularly the orbitofrontal cortex (OFC) and prefrontal cortex.

  • Damage to these areas (via lesions, stroke, or traumatic brain injuries like that sustained by Phineas Gage) produces distinct cognitive pathologies.

• Errors of Perseveration:

  • Motor Perseveration Tasks:

    • When a patient with frontal lobe damage is asked to copy an alternating rhythmic scribble pattern produced by a clinician, the patient initiates the motor action successfully but cannot apply cognitive and motor "brakes."

    • They continue generating repetitive loops across the paper, unable to disengage from the ongoing motor program.

  • Wisconsin Card Sorting Test (WCST):

    • Procedure: A deck of cards bearing diverse symbols varying in three dimensions—color, geometric shape, and numerical quantity—must be sorted under four stimulus card piles.

    • The subject is not told the sorting dimension (e.g., sort by color); they must deduce the active rule purely through experimenter feedback ("correct" or "incorrect").

    • The Rule Shift: Once the subject masters the rule (e.g., sorting by color for several consecutive trials), the experimenter silently changes the rule (e.g., shifting the rule to shape) without informing the subject.

    • Healthy Adult Response: Healthy individuals encounter unexpected negative feedback, experience momentary frustration, rapidly suppress the previous sorting rule, hypothesize alternative rules, and transition to sorting by shape.

    • Frontal Lobe Lesion Response: Patients with frontal damage pick up the first rule without difficulty. When the rule shifts, they receive persistent negative feedback ("incorrect"), yet they systematically continue to sort cards according to the old, unrewarded rule.

    • They are cognitively incapable of switching mental sets, representing a severe error of cognitive perseveration.

  • Developmental Parallels in Early Childhood:

    • A typically developing five-year-old child displays cognitive behaviors nearly identical to an adult with frontal lobe lesions.

    • The human prefrontal cortex is one of the latest brain structures to reach myelination and structural maturity.

    • If a five-year-old child plays the card sorting game by number, and the experimenter explicitly states, "Now we are no longer playing the number game; now we are playing the shape game," the child will verbally acknowledge the instruction, yet immediately sort the next card according to the original number rule.

• Absurd Cognitive Estimation in Frontal Damage:

  • While healthy adults leverage implicit heuristics to maintain rough ballpark accuracy, patients with frontal lobe lesions make wildly inaccurate, absurd numerical estimations when queried on real-world facts:

    • Average length of a human spine: Normal subjects guess within a realistic range ($2$ to $3\,\text{feet}$); frontal patients may estimate $12\,\text{feet}$.

    • Galloping speed of a racehorse: Frontal patients may estimate $2,000\,\text{mph}$.

    • Largest object normally found in a house: Frontal patients may suggest a "marble."

    • Camel population in the Netherlands: Frontal patients may estimate $4,000,000$ camels.

    • Cost of an ordinary toaster: Frontal patients may estimate $1,000$.

  • These errors highlight the frontal lobe's critical role in semantic verification, reality checking, and plausible estimation bounds.

• Goal-Directed Strategic Planning: The Tower of London Task:

  • Setup: A wooden apparatus with three vertical pegs of differing heights holding colored beads. The task requires rearranging beads from a starting configuration into a target goal configuration following specific physical constraints (e.g., moving one bead at a time).

  • Intact Frontal Function: Healthy participants engage in forward-looking mental simulation, planning out the required sequence of moves (e.g., a $5$-move mental sequence) prior to executing physical bead movements.

  • Frontal Pathology: Patients with frontal lobe damage fail to organize prospective goal paths; they immediately begin making trial-and-error, random movements without strategic foresight.

• The Somatosensory Disconnection Hypothesis: Emotion and Decision-Making:

  • Autonomic Skin Conductance: In healthy individuals, autonomic arousal can be quantified via Galvanic Skin Response (GSR), which measures changes in electrodermal activity driven by sweat gland activation during emotional or risky states.

  • The Orbitofrontal Disconnect:

    • Viewing emotionally salient stimuli (e.g., a photograph of one's own mother) produces an immediate elevation in GSR in neurotypical subjects, but fails to evoke an autonomic response in individuals with prosopagnosia (who cannot visually identify faces) or patients with specific orbitofrontal lesions.

    • Damage to the orbitofrontal cortex breaks the normal communication channel between cognitive thoughts and emotional somatic states.

  • Iowa Gambling Task (Card Deck Risk Paradigm):

    • Subjects draw cards sequentially from two distinct piles:

      • Deck A (Low-Risk/Advantageous): Provides consistent, modest rewards ($+50$ points/dollars) with occasional small penalties ($-100$ points/dollars). Net outcome over time is positive.

      • Deck B (High-Risk/Disadvantageous): Provides high immediate rewards ($+100$ points/dollars) with catastrophic occasional penalties ($-1,250$ points/dollars). Net outcome over time is devastatingly negative.

    • Normal Control Behavioral and Physiological Profile: Normal subjects initially sample both decks. Over time, they implicitly recognize that Deck B is toxic and switch exclusively to Deck A. Crucially, before executing a risky draw from Deck B, healthy participants exhibit anticipatory GSR elevations (sweating); their autonomic nervous system issues an emotional, somatic warning signal prior to conscious awareness.

    • Orbitofrontal Lesion Profile: Frontal patients fixate purely on the immediate $+100$ reward and repeatedly draw from Deck B, bankrupting themselves. They fail to generate anticipatory GSR arousal when hovering over the dangerous deck. Without the somatic warning signal to guide rational behavior, risk evaluation collapses.

Questions and Discussion

• Student Question regarding Examination Content and Brain Teasers:

  • Question: Will students be required to solve novel brain teasers, puzzles, or logic problems on the examination, or will questions focus on the specific concepts and experiments discussed during lecture?

  • Instructor Response: Students will not be asked to solve novel brain teasers or logic puzzles on the exam. Exam questions will present descriptive scenarios and lecture-based examples illustrating specific behaviors, and students will be expected to correctly identify and explain the corresponding heuristic, cognitive bias, or neuropsychological phenomenon (e.g., identifying when a scenario represents representativeness versus availability or framing effects).