Problem Solving and Situated Cognition

Murat Perit CakirCOGS 503

Outline

• Classical Theory of Problem Solving• Critiques to the Classical Theory• Situated Cognition Perspective– A case study of collaborative problem solving

• Discussion

Problem Solving

• We solve problems daily– Not necessarily limited to math or science

• Usually motivated by our needs/desires• Directed towards attaining a goal

• Problem solving research– aims to develop a scientific theory to describe the

main elements and dynamics of problem solving activities

Problem Solving

Current State Goal

Classical Information Processing Theory of Problem Solving

• Newell, A. & Simon, H. (1972). Human Problem Solving. Englewood Cliffs, NJ: Prentice-Hall.

• Very influential on AI, Decision Science• Based on well-defined, knowledge-lean problems– e.g. games and puzzles like chess, towers of hanoi

• Assumption– A theory for well-defined problems may be

augmented to cover ill-defined ones

Classical Theory

• Important concepts– Task Environment– Problem Space– States– Operators– Goals, subgoals– Heuristics

Task Environment• Abstract structure that corresponds to a problem

– Specifies the fundamental structure of the problem– Capacities of agents for action may bring different task environments to them– Task environment includes only those actions that either bring agents closer

or farther from the the goal condition

• Abstract-> Same task environment can be instantiated in different ways– Chess with physical pieces vs computer-based– Scratching your head, self-talk, simulating moves with gestures are all

deemed irrelevant to task performance

• Provides an interpretive framework– What counts as a relevant pb solving move

Task Environment

Elements of Problem Solving

• Goal directedness – behavior is organized toward a goal.

• Subgoal decomposition – the original goal can be broken into subtasks or subgoals.

• Operator application – the solution to the overall problem is a sequence of known operators (actions to change the situation).

Problem Space

• Problem space – the various states of the problem.

• State – a representation of the problem in some degree of solution.– Initial state – the initial (starting) situation.– Goal state – the desired ending situation.– Intermediate states – states on the way to the

goal.

Search

• Operator – an action that will transform the current problem state into another problem state.

• The problem space is a maze of states.• Operators provide paths through the maze – ways of moving through states.

• Problem solving is a search for the appropriate path through the maze.– Search trees – describe possible paths.

Production Systems

• Production rules – rules for solving a problem.• A production rule consists of:– Goal– Application tests– An action

• Typically written as if-then statements.– Condition – the “if” part, goal and tests.– Action – the “then” part, actions to do.

Features of Production Rules

• Conditionality –a condition describes when a rule applies and specifies action.

• Modularity – overall problem-solving is broken down into one production rule per operator.

• Goal factoring – each production rule is relevant to a particular goal (or subgoal).

• Abstractness – rules apply to a defined class of situations.

Sample Production Rules

Operator Selection

• How do we know what action to take to solve a problem?

• Possible criteria for operator selection:– Backup avoidance – don’t do anything that would

undo the existing state.– Difference reduction – do whatever helps most to

reduce the distance to the goal.– Means-end analysis – figure out what is needed to

reach the goal and make that a goal

Backup Avoidance

To solve each of these problems one must backup but most people will not do this and so have difficulty.

Tower of Hanoi

Missionaries and CannibalsMove 3 missionaries & cannibals across river. Cannibals cannot outnumber missionaries or else they will eat missionaries

Difference Reduction

• Select the operator that will produce a state that is closer to the goal state.– Or the one that produces a state that looks more

similar to the goal state.• Also called “hill climbing”.• Only considers whether next step is an

improvement, not overall plan.• Sometimes the solution requires going against

similarity – hobbits & orcs.

Means-End Analysis

• Newell & Simon – General Problem Solver (GPS).– A more sophisticated version of difference reduction.– What do you need, what have you got, how can you get

what you need?– Focus is on enabling blocked operators, not abandoning

them.– Larger goals broken into subgoals.

General Problem Solver

Summary of Information Processing Framework

• According to Newell & Simon PB Solving – The ability to reduce difference between current state & goal

state– Constrained by information processing system

• limited processing resources provide constraints on the degree to which multiple moves can be considered

• Assumptions underlying GPS’s design– Serial processing: execute one thing at a time– Limited working memory– Propositions are the basic unit of LTM– Heuristically or strategically driven process

GPS vs Human Problem Solvers

• Think aloud protocols conducted with human problem solvers show that humans approach puzzles in similar ways (Greeno, 1974)

• GPS sometimes deviate from human problem solving since humans tend to employ heuristics that will take them closer to a solution (hill climbing)

• GPS is resilient to cases when hill-climbing performs poorly (e.g. cannibals missionaries problem)

• GPS sometimes fail to find a solution since it applies means-ends analysis very rigidly

GPS vs Human Problem Solvers

Well vs Ill-Defined Problems• Puzzles

– unfamiliar– involve no prior

knowledge– all necessary info. is

present in the problem statement

– requirements are unambiguous

• Real-world problems– familiar– require prior knowledge– necessary information often

absent– solver must ask ‘what is the

goal’?

Case study of group problem solving

An Excerpt from VMT Spring Fest• A team of 3 upper-middle school students (14-16 years old)

• Students were recruited via their teachers, who are Math Forum users

• 5 teams completed 4 online sessions in 2 weeks

• A VMT project member was present in the room in case of technical difficulties

• The members of the most collaborative team were awarded with iPods

• The excerpt is taken from the first session of the team

Explicitreferencefrom chatto whiteboard

Activity awareness messages

Whiteboardhistoryscrollbar

Messageto messagereferencing

Drawingactivitymarkersembeddedin chat

Extra tabs (summary, math topic, wiki, browser, help manual)

Whiteboarddrawing controls

List of activeusers in the chat room

VMT Chat

137

Who is active on which tab

Task Description

1 1 4

2 3 10

3 6 18

4 10 28

N Squares Sticks

Here are the first few examples of a particular pattern, which is made using sticks to form connected squares:

How many squares will be in the Nth example of the pattern?How many sticks will be required to make the Nth example?

Task Description (cont.)Mathematicians do not just solve other people's problems, they also explore little worlds of patterns that they define and find interesting. Think about other mathematical problems related to the problem with the sticks.

Go to the VMT Wiki and share the most interesting mathproblems that your group chose to work on.

N=1 N=2 N=3

Excerpt 1

Co-construction of a new stick pattern

Excerpt 2

Constituting a shared problem

Excerpt 2 (cont.)

Developing a systematic counting approach

Questions

• How would you characterize the – Task environment?– Problem space?

• Where is the problem space located?• Were the group members primarily engaged in

search? • What is the role of representations in the

group’s work?• Do you think they understand each other?

Critique to Classical Theory from a Situated Cognition Perspective

• Framing & Registration• Interactivity & Epistemic Activity– Interactions with others and cultural artifacts– Role of external representations– Adding structure to the environment

• Socio-cultural context– Resources and Scaffolds– Knowledge rich

Framing

• Process of posing the problem in well-defined terms – i.e. constructing the graph structure, identifying initial and goal states, subgoals etc.

• Inappropriate abstraction filters away important cognitive processes relevant to problem solving– e.g. tic tac toe and the game of 15 are isomorphic mathematically but we rely on

different practices of reasoning when we play each

• Street math vs school math– Coconut sellers in Brasil, milk men in the US, grocery shoppers optimize their pb

solving performance by recognizing common patterns and using cultural resources

• How agents frame a problem, how they project meaning into a situation, determines the resources they see as relevant to its solution

• A psychological theory of pb solving needs to explain many phases and dynamics– how one sees a problem, why one sees it that way, how one exploits resources,

interacts with them, and solve the problems in acceptable time

Registration• The activity of selecting environmental anchors to tie mental/physical

representations to the world

• In ecologically realistic problem solving settings registration is non-trivial

• e.g. driving around in a new city with a navigator– You need to constantly anchor your physical location to the dynamic

representation presented in the navigator

• Registration is less of a problem in classical theory– Puzzles constrain pb solving interactions to occur in a spatially bounded

location• e.g. chess board, hanoi towers/pegs

Framing and Registration• Framing and Registration mutually inform each other

• Cooking example– Back and forth between the recipe and materials in the kitchen– Recipe frames the problem in terms of things that are relevant to the cooking

process• Ingredients, flame size, pots and pans, measurement cups• Not in terms of chemical reactions that took place during cooking

– Framing constrains actions

• Our understanding of problems is usually tied to the resources and tools at hand

• Problem solving involves moves back and forth between the abstract and the concrete

Role of External Representations• Problem solving is a process located partly in the mind, partly in the world• External representations have a key mediating role in problem solving• They bring affordances (cues and constraints on actions) that shape our

understanding of the problem• Later work of Simon and Larken (1987) attempted to incorporate external

representations in their classical model– But the focus remained on problem space and search heuristics, external

representations (diagrams, alternative symbolisms) were treated as secondary aids

Role of External Representations• Is an external representation same as its internal counterpart?• Experiments on mental imagery of ambiguous objects indicate that how

people visually explore an external and mental image may differ • What if some of the mental constructs we use during problem solving

have a similar property?

Further Criticism• Interactivity and epistemic activity

– Real world details of problem solving is not adequately captured by the notion of task space

– Some of the ignored actions may be important in understanding human problem solving (e.g. use of gestures, artifacts used to aid reasoning)

– Interactions with artifacts and other people can be used to explore the structure of a problem and to manage its complexity

• Scaffolds, practices, resources available as aids for problem solving– Problem solvers rarely work in isolation

• Knowledge-rich problem solving– Most problems are ill defined, understanding a problem requires background

knowledge– Even mundane tasks like shopping or cooking require background knowledge

But…

• Situated cognition does not offer an alternative theory of problem solving

• It offers a conceptual framework • Focuses on practices of problem solving, what

makes symbols, operators etc. meaningful to humans

Further Analysis of VMT Excerpts• Recurrent practical concerns for VMT participants

w.r.t. math artifacts and media affordances

– Identify and produce relevant mathematical artifacts to constitute a shared problem

– Refer to those artifacts and their relevant features

– Manipulate and observe the manipulation of those artifacts based on math practices known to participants

Representational Practices

• Group members display their reasoning by enacting representational affordances of VMT

– The drawing actions performed by 137 and Qwertyuiop• The organization of the lines revealed in 137’s first attempt led

Qwertyuiop to project what is needed

– Jason’s question with the explicit reference• Displays his understanding of the hexagonal pattern being developed

• Availability of the production process– Whiteboard affords an animated evolution of its contents that

makes the reasoning embodied in drawing actions visible

Referential Practices

• Group members establish relevancies across semiotic modalities by enacting referential uses of the available system features

– Verbal and explicit references• The indexical “hexagonal array” refers to shared drawing co-constructed on

the whiteboard. • Jason’s use of the referencing tool to highlight a particular stage

– Temporal organization of actions • The addition of 3 red lines were interpreted as a proposal to split the

hexagon into 6 parts, because it was made relevant in chat

Referential Practices (cont.)

• Through referential practices group members

– Isolate objects in the shared visual field and associate them with local terminology stated in chat

– Establish sequential organization among actions performed in chat and whiteboard spaces

so it has at least 6 triangles? in this, for instance

Shared Mathematical Understanding• In short, mathematical understanding at the group level is

achieved through the organization of representational and referential practices

• Persistent whiteboard objects and prior chat messages form a shared indexical ground for the group

• A new contribution… – is shaped by the indexical ground

• i.e., interpreted in relation to relevant features of the shared visual field and in response to prior actions

– reflexively shape the indexical ground • i.e., give further specificity to prior contents

– set up relevant courses of action to be pursued next

Summary

• Shared mathematical understanding is a process, a temporal course of work in the actual indexical detail of its practical actions, rather than a process hidden in the minds of the group members

• Mathematical understanding can be located in the practices of collective multimodal reasoning displayed by teams of students through the sequential and spatial organization of their actions