Problem Solving

Question 1

Defining Problem Solving

Answer 1

• A problem can be defined as a question proposed for solution
• This definition is brief and framed in the very specific context of cicumsribed domain e.g. mathematics
• A problem can also be viewed as any situation in which a person is confronted with a dilemma or a choice or an uncertainty in an action or belief
• A question to be answered in which the process toward a solution is known by the person is not a problem in the sense that there is no difficulty in enacting a solution e.g. solving a maths problem
• Problem solving is also a very important issue when building AI systems or developing expert systems for helping professions e.g. Medical Expert Systems involved in aiding diagnosis of illness

Question 2

Characteristics of Problem Solving

Answer 2

• In studying problems researchers have identified the concept of a problem into three central elements:
  
  • I – Initial state
  • Ii – goal state
  • Iii – steps to goal state
• The number of steps from I -> ii may vary from few too many (in order of thousands in the case of computer programs).
• The existence of a “problem” is captured in the statement that the steps number or type is NOT obvious i.e. known to the solver.
• If the steps are known, especially to complex problems we are in the realms of expertise, which will be commented on later.

Question 3

Tower of Hanoi Problem

Answer 3

• Is the problem well or ill-defined?
  
  • How many paths are there to the minimum move solution?
  • How much knowledge must be brought to bear to solve problem?
• Is the problem Knowledge Rich or Lean?
  
  • How much knowledge of the domain is needed to solve the problem?
  • Is specialised knowledge required?
• Is the method of solution strong or weak?
  
  • Is there a clear well known method of solution?
  • What method did you use?
• Notice that the 5^th move is essentially backwards away from the goal, this is often a difficult part of the task for many subjects.

Question 4

The Problem Space

Answer 4

• A problem space consists of three elements
  
  • Symbolic structures
  • A set of operators i.e. solution actions that together create a solution path
  • A set of constraints
• The emphasis is on the search through the problem space or a solution path
• Newell & Simon drastically changed the psychological study of problem solving by introducing the concept of the interaction of the cognitive architecture of the human mind and the task environment (the problem at hand)
• The concept of searching the space is very important to such issues as working memory and connectionists model of LTM

Question 5

Means-Ends Analysis

Answer 5

• M-E analysis is heuristic method
• M-E analysis involves the following steps
  
  • Determining what the goal state is
  • Assessing the ‘distance’ between the current problem-solving state and the desired goal-state.
  • Choosing an operator for reducing the greatest difference between the states.
• The critical operator in this process is the use of sub-goals. An operator is applied and when an obstacle is encountered the problem solver sets the sub-goal of removing the obstacle.
• This process is iterative and is repeated until the solution is attained.
• This assumes a linear view of the problem state and connections between elements of the problem situation.
• Many real world complex problems are characterised by interconnections between elements of the problem such that when one aspect is solved that alters the problem so that the goal state may not be possible with the general solution path that has been taken.

Question 6

Problem representation - Insight Problem

Answer 6

• These are problems that once solved are characterised by a sudden comprehension of the solution or “aha”
• An example is western style humour
• These types of problems illustrate the situation of the role of associative memory.
• One of the barriers to solving insight type problems is the inability of the problem solver to loosen their commitment to a particular train of thought.
• The process of de-centering and coming at problems from different ‘angles’ demonstrates the role of LTM and a type of preservation
• Often ‘sleeping on’ problems is a method of relinquishing previous narrow thinking.
• These problems illustrate the central role of constraint representation
• The representation of the initial and goal states are often easier than the constraints which then divert or even barrier a productive path.

Question 7

Problem representation - impedances to accurate problem representation

Answer 7

• The factors or elements of a problem that have been found to make an accurate representation possible are [Sternberg & Ben-Zeev, 2001, p153]:
  
  • The degree of novelty of the given entities and relations (presenting people with unfamiliar objects),
  • The extent to which rules are counter-intuitive, and
  • Whether the content is abstract and therefore difficult to visualise
• This is relevant to the material on the condition reasoning problem of Deontic reasoning where subjects do much better with experienced based content (e.g. Drinking alcohol laws) rather than abstract rules.

Question 8

Problem representation

Answer 8

• The concept of ‘one track’ thinking is where the problem solver is seeming caught on one ‘line’ of thought.
• This fixedness can be the result of past experience in the type of problem or general experiences such as success in problem solving with a particular set of methods, e.g. applying maths to romance.

Question 9

Model Generation in Problem Solving

Answer 9

• The theory of mental models is the work of Johnson-Laird.
• First step in problem solving is the generation of a model or mental representation.
• The use of diagrams is central to this process and is seen as a way of representing the relevant information from the problem statement.
• In the model or image of the problem the representation of the constraints becomes a critical issue.
• The model is a device for collecting the relevant information and therefore acts as a basis for understanding the context, operators and end states of the problem.
• Each thought element can be viewed as a knowledge state.
• Diagrams and other ‘external’ representations are aids for working memory which is the most critical aspect of the process for many problems

Question 10

Theories of Problem solving -

Newell and Simon (1972) Model

Answer 10

• This theory treats people as information processing systems and so was foundational in the early work on AI.
• The emphasis in on the interaction between the task environment and the problem solver (be it human or computer).
• This theory took the view that people or machines take units of information and transform them by ‘rules’ that include criteria for grouping symbols (semantics), retrieval of stored information (LTM) and procedures for organising the information within the context of the problem.
• The theory also brought into focus the limitations of the Human Information Processing (HIP) System
  
  • How much information it can keep in working memory
  • Information encoding limitations
  • Fidelity and amount of stored information
  • Information retrieval
  • Maintenance of motivation and arousal
• Laird and Rosenbloom (1996) note that Newell sums up the HIP system the actions it intends are those that will achieve its goal. That is the system is adaptive and goal directed not irrational.

Question 11

Theories of Problem solving -

Ohlsson’s Theory

Answer 11

• Reading a problem generates:
  
  • The problem’s givens
  • Some solution criteria
• Based in this mental representation the solver accesses a set of mental operators which the solver applies to the problem in a ‘tentative’ manner.
• Problem solving is sequential with applying one operator at a time (a linear process)
• Retrieval of relevant operators occurs through spreading activation i.e. bits of information in the semantic network are differentially activated and recalled according to their activation level. This in an unconscious process.
• The mental representation acts as a memory probe via semantic relationships. No semantic relationships no retrieval.
• According to Ohlsson Theory there are three ways to change the mental representation;
  
  • Elaboration : the problem is expanded in terms of semantic association or further examination of the problem statement or analysis of the constraints
  • Re-encoding : Objects or operators are reviewed or reclassified e.g. pliers as a pendulum

Constraint relaxation : are the constraints in the problem or the thinking of the problem solver e.g. the three line, nine circle problem

Question 12

Development of Expertise

Answer 12

• When compared to Novices, experts are characterised by (Sternberg & Ben-Zeev,2001,p303):
  
  • Have large, rich schemas full of declarative knowledge about the domain
  • Have well-organised, highly interconnected units of knowledge in schemas
  • Spend high proportions of time representing a problem rather than trying ‘solutions’
  • Develop representations based on structural (deep) similarities among problems
  • Work forward from given information to implement strategies to find unknown information
  • Generally chose a strategy based on existing elaborated schemas of problems solved in area
  • Carefully monitor own problem-solving strategies
  • Take more time over novel problem representation
  • Show flexibility in responding to contradictory information and modifying initial problem representations and initial strategy