Lecture 15: Problem Solving

Question 1

Problem solving definitions

Answer 1

• A multi-step process to shift your current problem state to a goal state (trying to move from a current problem state to a goal state)
  
  • Math problems
  • Social problems
  • And everything in between
• How do we engage in this process of problem solving efficiently?
• Involves a lot of processes: memory, attention, perception, decision making

Question 2

Engaging in problem solving is …

Answer 2

• Cyclical
  
  • Enact steps that occur in a loop (series of steps that we go through to reach a solution).
  • Step 1: know the problem/define the problem. Represent it mentally
  • Step 2: Think about how you could solve it (defining strategies to attack your problem)
  • Step 3: decide on the best solution
  • Step 4: implement best solution.
  • Step 5: evaluate what you did to see if you used the best strategy and reached a solution.
  • Store this cycle in memory and apply it to new problem.
• Recursive
  
  • Repeat this cycle as many times a necessary to find a solution
• Applicable
  
  • Apply successful cycles (solutions) to hew problems

Question 3

Types of problems

Answer 3

Well defined problems: Unambigious, straight path from problem to solution. Can apply existing rules or algorithms to reach a solution.

Ill defined problem: problem is ambiguious. Could be many solution paths or it can have many types of goals. Have to determine the context and the solution and bring more info to formulate the solution plan – need to take into account more information.

Question 4

Well-defined problems

Answer 4

• Goal directedness
• Problems with a defined goal state and set task constraints such that there are clear steps
• Ex: soduku

Question 5

Ill defined problems

Answer 5

• Ambiguous situations that:
  
  • Have few limitations (rules) for how to solve the problem
  • Must create your own path
  • Multiple solutions
  • Need to take into account a lot of other information
• Social problem solving is a form of ill defined problem solving:
  “I need to figure out how to make new friends.”
  “I want to learn a new musical instrument”

Question 6

Ill-defined problems carry a load

Answer 6

• This study controlled for content to look at differencess in how people solved well-defined and ill defined problems.
• They had to solve anagrams. Create a word from jumbled letters.
• They made one anagram more well-defined task (easier to solve the problem).
• To measure Ill defined - they just showed a jumble of words and said to make a word out of it.
• Greater activity in the right lateral prefrontal cortex for ill-defined anagrams (this region is important for organizing cognitive processes) .
• Solving ill-defined problems carries a greater ‘cognitive load’. With ill-defined problems there is more organization, more maintaining that you have to do with information (greater cognitive load).
• Cognitive load: the amount of information held in mind at one time

Question 7

Moravec’s paradox

Answer 7

• Artificial intelligence (AI) can solve well-defined problems (chess, go) well, but not ill-defined problems and simple skills
• “Everything that’s easy is hard, and everything that’s hard is easy”
• AI is often defined by the use of algorithms, deep neural networks, that work well with certainty but not with un- certainty
• AI is not very good at predicting future events, navigating unnfamiliar environments, riding a bike (skill based tasks).

Question 8

Problem solving algorithms

Answer 8

• Strategies to move through a “problem space”
• Often used to solve well-defined problems
• A problem space is a representation of a problem and algorithms that can take you to a solution. Within a problem space we have 4 main features:
  
  • Initial and goals states
  • Intermediate paths and operators (transformations that occur as you move from state to state): actions to change between states
  • Task constraints (allowable moves that can be made).

Question 9

The Tower of Hanoi

Answer 9

• Well defined problem
• Move 3 discs from peg A to C so they are in the same initial order
• Task constraints :No disc can lie on top of a smaller one & only one
  disc can be moved at a time (i.e., the top disc of a pile)

Question 10

The Tower of Hanoi problem space

Answer 10

1. Initial and goals states
2. Intermediate states and operators
3. Task constraints

You have to figure out how to move from the initial state to the goal sstate that you want by implementing the operators and moving between those intermediate stages. The operators that are going to get you to the desired goal are constrained by those rules you have.

Question 11

A brute force approach

Answer 11

• Brute force approach means that you consider all the entire problem space and every possible move to get to a solution. Consider everything
• Systematic algorithm that represents all the possible steps from the problem to goal state
• Guaranteed to find a solution, but inefficient (takes a lot of time)
• Combinatorial explosion: computing too many alternatives. System will shut down and not reach a solution.

Question 12

Narrowing down search: Heuristics

Answer 12

• Good problem solving often comes when we use heuristics or strategies to select the best operators to move effectively to a goal.
• Menal shortcuts that you take so that you dont have to consider all the information out there.
• Strategies to select moves in a problem space
• Helps avoid combinatorial explosion
  
  • Hill climbing strategy
  • Means end analysis

Question 13

Hill climbing strategy

Answer 13

• Hill climbing strategy is the difference reduction strategy.
• In hill climbing, you are going to move in any direction hthat is going to take you closer to that goal.
• Select the operation that brings you closer to the goal without examining the whole problem space.
• This strategy can lead to a false outcome, a ‘local maxima’ (subgoal) is mistaken as the final goal
• Does not always work because some problems require you to move away from the goal in order to solve it

Question 14

The hobbits and orcs problem
(Cannot be solved with hill climbing)

Answer 14

• Three Hobbits and Three Orcs are on one side of a river, and they all want to cross to the other side
• There is one boat that holds 1 or 2 creatures
• If there are ever MORE Orcs than Hobbits in one place, the Orcs eat the Hobbits.
• At least one creature must bring the boat back each time.
• How can you get everyone safely to the other side
• You must move away from the goal to get there
• IF you look at the soution on how you get these hobbits and orcs to the other ssidde with all these task constrainsts is that theres’s a lot of intermediate steps that you have to take. And in these steps a lot of the time you have to move away from the goal state.
• shows that

Question 15

Means ends strategy

Answer 15

• What “means” do I have to make the current state look like the goal state I want to be in? We want to reduce the difference between our current state and our gaol, butt we are going to consider the means that we have to do to get to that goal state (consider all the possible routes).
• Identifying sub-problems to complete the goal
• Includes forward and backward movements and constantly evaluating the difference between current and goal states
• More flexible approach than hill-climbing
• Takes into account the full problem space.

Question 16

Means end: Meet the Cog Dog at the park

Answer 16

• Advantage of this straegy is that you are able to set up these new goals or as you move through the problem space, you are considering where you are in the problem space.
• Current state: home alone
• Goal: being with the Cog Dog at the Park
• Evaluate: there are two major differences
  (subproblems) Location and Company
• Pick the most relevant difference (Location): the one that will get you closer to your goal first. You are going to try to solve that subgoal, moves towards the intermediate step that will involve changing location
• Engage operators (Walk to the Park)
• Evaluate …

Question 17

Expertise (familiarity) and problem solving

Answer 17

• Experts are more familiar with certain information and so they represent a problem differently than non-experts
• Experts will spend much more time defining a problem and a problem space then novices or non experts/ novices that will just try to run towards a solution.
• Experts will take a more holistic or globabl view of a problem space rather than getting drowned by the detail.
• Expert radiologists use ‘global’ visual processes when viewing scans (study used eye tracking to examine how expert and novice radiologist would view chest X-rays).
• novices used a lot of eye fixation and saccades. Experts had fewer sort of eye fixations so they weren’t sort of stoping and looking at details. Shows that as we become more familiar, we can implement more search strategies.

Question 18

Analogical problem solving

Answer 18

• Making comparisons between two situations; applying the solution from one of the situations to the other situation
• ** Target**: the situations the person is currently in (The Tumor Problem). The one you are trying to solve.
• Source: the situation that shares similarity with the target (the Fortress Story)
• Consider or remember another problem that you solved that is similar in some way to a current one. It can have similarities in the underlying structure or general meaing of the past experience that you are applying to a current one.
• Analogical problem solving is very effective for illdefined scenarios.

Question 19

Analogical Problem Solving

Answer 19

• People aren’t very good at using analogies unless they are ”reminded”
• Without the hint, a person must look beyond surface details, and consider the general structure (the gist)
• Most success when a source and target share surface and structure
• E.g., Using a past school-related problem to solve a current school-related problem vs a current relationship-related problem

Question 20

The Einstellung effect

Answer 20

• The bias to use familiar methods to solve a problem. To overlook more efficient or althernate methods whenever problem solving
• “I always do it this way ..” –> rigit thinking
• Can result in an inability to seek and use a better method to solve a given problem
• Leads to rigid thinking and blocks in problem solving. Leads you to less optimal solutions
• Functional and Mental Fixedness

Question 21

Functional fixedness

Answer 21

• The inability to see beyond the most common use of a particular object
• “Fixed” on the function of an object that you know.
• HArd time recognizing that you can use something in your environment a different way to get you to a solution.

Question 22

Classic test of functional fixedness

Answer 22

• Only 39% of people can find the solution within 10 minutes
• Hints help. Prior to task, swinging arms in a solution relevant manner led to the solution

Question 23

The candle problem

Answer 23

• Researchers give the participantss a candle , a thumbtack and a box of matches.
• They are asked to use these objects to attach the candle onto the wall so it didnt drip wax onto the table.
• BEst way is to tack the matchbox onto the wall to use it as a platform.
• Few people realized to do this because they were fixated on the funnctions they knew od the objects. The matchbox was used to light the cadle.

Question 24

The development of functional fixedness

Answer 24

• Children of different ages solved the ‘candle’ problem
• Difference with respect to pre- utilization: experience with the objects
• No fixedness in children without pre-utilization. Faster reaction time for children without pre-utilization. Children are more resistant to functional fixedness because they have less experience with those objects.
• Too much experience may lead to fixedness and the Einstellung effect.

Question 25

Mental fixedness: Overusing mental sets

Answer 25

* Responding with previously learned rule sequences even when they are inappropriate or less productive * The tendency to respond inflexibly to a particular type of problem and not alter your response. Can be negative because you might not be able to see a more productive solution. * Mental sets = what you are going to pay attention to in a certai situation or how you are goinng to solve something. * This is also the Einstellung Effect because it showss that we have a tendency to respond inflexibily to a particular type of problem.

Question 26

Mental sets: The water jug problem | Example of mental fixedness

Answer 26

* In the water jug task, you are told how much water 3 jugs can hold. * You are given a desired amount of watter and you have to use the capacity of the three jugs to get that specific solution amount. * You can use that formula to get to all of these desired amounts across the 8 problems.

Question 27

Mental sets: The water jug problem

Answer 27

* Once you get to the later solutions, problems 7 and 8, there's a much simpler way to reach the solution. * If people are given the water drug problems in order (from 1-8) , you find that a lot of people will implement that less effective solution for problem 7 and 8. * But an experimental group, if you give them just problem 7 and 8 then they will use the simpler task.

Question 28

Insight

Answer 28

* A productive thinking process of forming new patterns or ways to view a problem * Restructuring a problem in a new way leads to a sudden solution (or viewing a problem in a new matter) * The Aha moment (often these solutions seem to come quite quickly) * Gestalt switches: the experience of having a sudden switch in how you see something. We perceive things in a whole (in a all or none fashion) - restructure problems just like you can restructure bistable figures.

Question 29

Four features of insight

Answer 29

* Suddenness: The solution pops into mind with surprise * Ease: The solution comes quickly and fluently * Positive: A pleasant experience, even before assessing if the solution is effective * Confidence: The solution is believed to be the right one

Question 30

Insight: The triangle problem

Answer 30

The triangle shown below points to the top of the page. How can you move three circles to get the triangle to point to the bottom of the page? * Problems that are thought to be best solved with insight (you have to restructure the problem).

Question 31

Insight problems: The necklace

Answer 31

* You are given four separate chains that are each three links in length * It costs 2 cents to open a link and 3 cents to close a link. * All links are closed at the start. * Join all links of chain into a single circle at a cost of no more than 15 cents. * Open each link of one of the chains (2 cents * 3 = 6 cents) * Attach the other chains together by closing these links (3 cents *3 = 9) * You are restructuring the problem, a key characteristic of insight * You have to realize that you dont need to use each chain as a whole - see this problem and pieces in a new way.

Question 32

Insight results from impasse

Answer 32

* mental impasse is being stuck in a solution path * leads to sudden insight from restructuring the problem to see a new solution * We try to solve the problem the way we usually do but then we reach a block and we realize we're not going to actually reach the solution taking our normal way so we have this block. A lot of research thinks that these impasses are necessary for insight.

Question 33

Insight and the subjective experience

Answer 33

* Insight problem solving feels like it happens suddenly * People cannot accurately predict performance (finding solution) * Non-insight problem solving comes with awareness, ease and pleasure. * Step by step algorithms help predict performance

Question 34

Insight and the subjective experience

Answer 34

* Participants solved ten problems: * Five verbal insight problems * Five non-insight algebra problems * Made ‘warmth ratings’ (how close they felt they were getting to a silution) every 15 seconds * A person’s feeling that they are approaching solution * Warmth ratings predicted performance on algebra by not insight problems * Steady increase for warmth problems but for the insight problemss they weren't associated with how peiople were to the actual solution. People arent able to judge how close they are.

Question 35

Insight is defined by the experience

Answer 35

* Participants shown problems and asked to rate feeling of knowing ( “Do you feel that you will know the answer to this problem?”) * Completed algebra and insight problems * Feeling of knowing ratings predicted algebra but not insight problem solving ability * Metacognitive assessments (What you know about what you know) is not accurate for insight problems