# Problem Solving and Situated

Τεχνίτη Νοημοσύνη και Ρομποτική

23 Φεβ 2014 (πριν από 7 χρόνια και 6 μήνες)

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Problem Solving and Situated
Cognition

Murat
Perit

Cakir

COGS 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

a
ims 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.
g
ames 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

Problem Space

States

Operators

Goals,
subgoals

Heuristics

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

self
-
talk, simulating moves with gestures are all

Provides an interpretive framework

What counts as a relevant
pb

solving move

Elements of Problem Solving

Goal directedness

behavior is organized
toward a goal.

Subgoal

decomposition

the original goal can
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 Cannibals

Move 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
-
D
efined 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

Explicit

reference

from chat

to white

board

Activity
awareness
messages

Whiteboard

history

scrollbar

Message

to message

referencing

Drawing

activity

markers

embedded

in chat

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

Whiteboard

drawing

controls

List of active

users in the

chat room

VMT Chat

137

Who is active

on which tab

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?

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 math

problems 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

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

Socio
-
cultural context

Resources and Scaffolds

Knowledge rich

F
raming

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
B
rasil
, 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

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

M
athematical 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