Thinking and Intelligence

Thinking and Intelligence

Problem solving • What is a problem? • A situation in which there is a desired goal, but it is not clear how to reach the goal

Parts of problems • initial state • the situation at the beginning • goal state • the desired situation • allowable operations • things you can do to move from one state to another • problem space • all the the possible states that you can be in

Parts of problems e.g. Tower of Hanoi puzzle

Parts of problems • Initial state • the disks on peg A (as shown) • Goal state • the disks on peg C • Allowable operations • only move one disk at a time • cannot put a larger disk on a smaller disk • Problem space • the 18 different arrangements of the disks on the 3 pegs that can occur by following the rules

Problem space for anagrams • unscramble these letters to make a word: G D I • initial state: G D I • goal state: a word • allowable operations: switching the order of the letters • Problem space is small (only 6 possible states): GDI GID DGI DIG IDG IGD

Larger problem spaces • Unscramble these letters to make a word: GSHOLYCOPY • there are 907,200 possible arrangements • Your knowledge of language allows you to constrain (restrict) the problem space • reducing the search space to a manageable size. • In general, problems with larger problem spaces are harder to solve

Types of problem • well-defined problems • have clear specifications of the initial state, goal state, and the operations for reaching the goal state • e.g., anagrams, math puzzles, chess puzzles, etc • ill-defined problems • lack clear specification of the start state, goal state, or the operations for reaching the goal state • e.g., how do I become rich and famous? • e.g., how can I convince my parents to let me travel to Europe this summer?

Blocks to Problem Solving • Interpretation blocks • Problem solving errors often arise from misinterpreting the problem • e.g., If a bat and a ball together cost $1.10, and the bat costs $1 more than the ball, how much does the ball cost? • many people misinterpret this question as requiring a simple subtraction of the numbers given in the question and answer “10 cents”

Interpretation blocks • If bob can cut the lawn in 5 hours, and bill can do it in 3 hours, how long will it take for them to do it together? • it is easy to misinterpret this this question as requiring a simple average of the two times to get “4 hours” • In these examples many people tend to get fixated on one (incorrect) way to interpret the problem, and find it hard to consider other interpretations • Fixation is the inability to create a new interpretation of a problem

Fixation example • Connect these nine dots with four straight lines, all connected. Do not lift your pen

Fixation example • Typically, we fixate on the assumption that the lines have to be within the boundaries of the 9 dots. • this interpretation restricts the solutions we can think of • Once we change our interpretation of the problem, the solution becomes easier • the trick is to let your lines go outside the box

Fixation example • Connect these nine dots with four straight lines, all connected. Do not lift your pen • insight • finding a new way to interpret a problem that quickly leads to the solution

Functional fixedness • We tend to see objects as having only the set of functions that they are typically used for • e.g., a pair of scissors • e.g., a frozen leg of lamb • Functional fixedness is the inability to see that an object can have a function other than its typical one • Limits our ability to solve problems that require using an object in a novel way • e.g., if you need a screwdriver but don’t have one, a dime could be used instead

Two-string problem

Two-string problem • Functional fixedness plays a role • Scissors seen as a cutting tool • Useless! The ropes need to be made longer, not shorter. • But the scissors could also make a convenient weight to turn the rope into a pendulum

Blocks to Problem Solving • Mental set • the tendency to use previously successful solution strategies without considering others that are more appropriate for the current problem

Water-jug study (Lurchins, 1942) • The participant is given a set of three jugs of various capacities • Told to measure out a desired quantity of water • e.g., Jug A: 21 oz Jug B: 127 oz Jug C: 3 oz Goal: 100 oz • Answer • Jug B – Jug A – Jug C – Jug C • i.e., B – A – 2C

Water-jug study

Water jug study • All the problems can be solved using B-A-2C • But problems 6 and 7 can be solved more quickly • #6 the best solution is A – C • #7 the best solution is A + C • Which solution will participants use for #6 and #7? • 83% of participants who did all the problems in order used B – A – 2C on on problems 6 and 7. • < 1% of participants used B – A – 2C if they had NOT been asked to previously solve problems 1-5.

Algorithms vs. Heuristics • You are in a town that you’ve never been to before. You are hungry and want to find a fast-food restaurant. • You could find a pay phone then look in the yellow pages for the address of a restaurant. • This would be using an algorithm • Or, you could drive to the free-way. Usually there are fast-food restaurants by the freeway exits. • This would be using a heuristic

Algorithms versus Heuristics • Algorithm • a predetermined procedure for solving a problem that guarantees a correct answer to that problem • Heuristic • a rule-of-thumb that, based on past experience with similar problems, will likely lead to an acceptable solution (but which does not guarantee a correct answer).

Heuristics • Many studies show that we use a variety of heuristics when solving problems and making decisions • anchoring and adjustment • Uses an initial estimate as an anchor and then this anchor is adjusted up or down • e.g., estimating how long it will take to drive home in the winter: “It took 4 hours to get here in August, but the roads may be icy now, so it will take a bit longer... about 4hrs and 15 minutes.”

Anchoring and adjustment • Tversky and Kahneman (1974) • groups of high school students given 5 seconds to estimate the answer to either: • 8  7  6  5  4  3  2  1 or • 1  2  3  4  5  6  7  8 • The students calculated the first few steps and then adjusted that value upwards to arrive at their estimate • the first group’s median estimate was 2,250 • the seconds group’s estimate was 512 • The first group start with larger values (anchor), and so their final estimate is higher • Neither group adjusted enough though. The correct answer is 40,320

Types of Heuristics • working backwards • attempting to solve a problem by working from the goal state backward to the start state • e.g. Water lilies growing in a pond double in area every day. On the first day of spring, only one lily pad is on the surface of the pond. Sixty days later, the entire pond is covered. On what day is the pond half covered?” • This can only be solved by working backward • e.g., when should I start working on my final term paper? • e.g., tracing the correct route through a maze

Types of Heuristics • means-ends analysis • breaking down the problem into sub-goals and working toward decreasing the distance to the goal state by achieving these sub-goals • e.g., If you lock your keys, wallet, phone etc., in your car. • goal: drive home • sub goal: get roommate to bring your spare keys to you from home • sub goal: call roommate • sub goal: find a quarter • sub goal: find a pay phone • etc.,

Thinking Under Uncertainty Judging Probability Cognitive Biases in Reasoning

Judging Probability • Two main heuristics we use to make judgments about probabilities... • The Representativeness Heuristic • The Availability Heuristic

The Representativeness Heuristic • A rule of thumb for judging the probability of membership in a category by how well an object resembles (i.e., is representative of) that category • The more representative the object is, the more probable it is that is belongs to that category • e.g., You hear about a person who likes to write, read, and interpret poetry. Is it more likely that this person is: A hockey fan or an English professor? • We tend to use the representativeness heuristic because the mind categorizes information automatically

The Representativeness Heuristic • This heuristic works well, and helps us make quick decisions about many complex issues. • Unfortunately, it sometimes leads us to make incorrect decisions. • e.g., Linda is 31, single, outspoken, and very bright. She majored in philosophy in college. As a student she was deeply concerned with discrimination and other social issues, and she participated in antinuclear demonstrations. • Which statement is more likely? • Linda is a bank teller • Linda is a bank teller and active in the feminist movement

The Representativeness Heuristic • Linda is much more likely to be a bank teller than to be a bank teller and be active in the feminist movement • The conjunction rule: the odds of two uncertain events occurring together is always less than the odds of ether occurring alone • e.g., there are lots of bank tellers, but only some of them are active feminists • However, our judgement is biased by the representativeness heuristic • the description of Linda has many properties that are representative of an active feminist • leads to the conjunction fallacy

Judgements of randomness • Which of these sequences of coin tosses is most likely to occur? • H H H H H H • H T T H T H • We expect random sequences to look like (i.e., be representative of) randomness • it should have about equal numbers of heads and tails • it should show no obvious order • Gamblers fallacy • The erroneous belief that a chance process is self-correcting in that an event that has not occurred for a while is more likely to occur

Availability Heuristic • When assessing the likelihood of an event we assume that • the more available an event is in our memory, the more probable it is in real life • the probability of an event is related to how frequently it has happened in the past, and events that occur more often are easier to recall • e.g., May be used to answer these questions • What type of pants do most students wear? • What drinks do most people have at breakfast?

Availability heuristic • This works well. • But there are other factors that also make events easier to recall • e.g., recent media coverage • e.g., vivid, exciting, scary things • Events that have been in the news recently, or dramatic vivid events tend to be estimated as being more likely. • e.g., The movie Jaws led people to overestimate the probability of being attacked by a shark

Examples illustrating the availability heuristic • K words: • there are many more words with 3rd letter K • but these are harder to think of (they are less available), so we typically respond there are more that start with K • Violent versus property crimes • only 13% of crimes were violent in (1995) • 1,798,790 violent crimes • 12,068,400 property crimes • media coverage is greater for violent crimes, so these are more available, so we over estimate the occurrence of violent crime

Examples illustrating the availability heuristic • Deaths per 100, Million • All accidents 55,000, strokes 102,000 • Electrocution 500, asthma 920 • Homicide 9200, diabetes 19,000 • Lightning 52, appendicitis 440 • Drowning 3600, leukemia 7100 • Typically we perceive the cause that is easiest to picture, or more publicized, as being more common

Cognitive biases in reasoning • Confirmation bias • Illusory correlations • Person-who reasoning • Loss aversion

Confirmation bias Wason selection task • Here are four cards • Each card has a number on one side and a letter on the other side • Which cards should you turn over in order to test the truth of this statement? “If a card shows a vowel on one face, then its opposite face shows an even number.” P E 4 7

Wason selection task • Most people choose to turn just the E card. • This allows you to confirm the hypothesis • if there’s a vowel (E), then there is an even number • BUT the correct answer is the turn the E card and the 7 card • checking the 7 is required to make sure there is not a vowel on the other side. • a vowel here would disconfirm the hypothesis • In general: we tend to only seek information to confirm our hypothesis • we ignore evidence that would refute our hypothesis

Confirmation bias • Can help us understand people’s beliefs in pseudoscientific ideas • e.g., • People who believe aliens are inhabiting the earth, tend to seek out information that confirms this • but they will not look for information that disproves it

Loss aversion • You have preordered a ticket for a movie. Once you get your ticket (which is non refundable) you hear that the movie is terrible, and that the movie theatre itself has uncomfortable seats. • Do you: a) go to the movie anyway? b) throw away your tickets and do something else more enjoyable?

Loss aversion • The cost of the ticket is lost and cannot be recovered. • it is called a sunk cost • This ought not come into consideration when deciding to see the movie. • Nevertheless, even though we know we will not enjoy the movie, we find it hard to ‘waste’ the cost of the ticket • even though we’d be better off doing something fun • Loss ‘aversion’ • has impact on economics, investing decisions, advertising, relationships, etc...

Intelligent Thinking Intelligence Tests Theories about Intelligence Nature versus Nurture

The first intelligence tests • Binet & Simon (early 20th century France) • Working on the problem of mental retardation when France switched to mass public education • Developed a test to diagnose children who were subnormal • Published in 1905, this test was the first accepted test of intelligence

Mental Age • Binet and Simon’s test was based on the concept of mental age • the age typically associated with a child’s level of performance • If a child’s mental age was less than his or her chronological age, the child would need remedial work

Terman • Lewis Terman at Stanford University revised Binet and Simon’s test for American school children • This test became known as the Stanford-Binet • Terman used an intelligence quotient (IQ) formula to report the scores on his test • IQ = 100  (mental age/chronological age) • If a child’s mental age as (assessed by the test) was greater than the child’s chronological age, the child’s IQ was greater than 100 • If a child’s mental age was less than the child’s chronological age, the child’s IQ was less than 100 • Note: This IQ formula is no longer used

Thinking and Intelligence