Educational Objectives
For Problem Solving
Skills
David Sands and Tina Overton
Dept. Physical Sciences
We need:
To identify problem-solving processes
Expert-novice comparisons
To teach these processes
Field dependence - independence
Classify these processes
A taxonomy - Marzano
Why a new taxonomy?
Bloom’s taxonomy is about 60 years old
Confuses knowledge with processes
“The new taxonomy avoids this … by postulating
three domains of knowledge that are operated on
by three systems of thought and their component
elements” – Marzano and Kendall, 2008
Thought systems: self system
metacognitive
cognitive – 4 components
Problem solving
“Whenever there is a gap between where you
are now and where you want to be, and you
don’t know how to find a way to cross that gap,
you have a problem.”
Hayes, J. The complete problem solver, 1980
“Problem solving is what you do when you don’t
know what to do.”
Wheatley, G H. Problem solving in school mathematics,
1984 (MEPS Technical Report, School of Mathematics
and Science Centre, Purdue University)
Can problem-solving be analyzed?
Up until the 1970s problem-solving was general
Around 1980 the importance of domainspecific knowledge was recognised.
Two very commonly cited papers:
Expert and Novice Performance in Solving Physics Problems, Jill
Larkin, John McDermott, Dorothea P Simon, Herbert A Simon,
Science 208 1980 1335-1342
Categorization and Representation of Physics Problems by Experts
and Novices, Micheline T H Chi, Paul J Feltovich, Robert Glaser,
Cognitive Science 5 1981 121-152
Larkin et al:
Computer modelling of problem-solving:
cognition is an analogue of a computer programme
PS framed in terms of productions resulting from condition-action pairs;
if x then do y.
Students - direct syntactic translations, experts - make representations
(surface features vs. deeper structure)
Pencil and paper representations augment working memory
Distinguish between internal and external representations
concentrate on internal
(Nersessian postulates that internal and external representations form a
coupled system which facilitate analogical reasoning)
Essential conclusion; novices use means-end analysis, experts use a forward
looking approach.
Chi, Feltovich and Glaser
asked novices and experts to categorize end-of-chapter physics
problems
novices concentrate on surface features, eg. Common components, such as
an inclined plane
experts concentrate on the deep structure, eg the physical principles behind
the problem, such as conservation of energy or momentum.
Production rules are also mentioned; experts work forward from principles,
novices search for a solution.
Clearly parallels the work of Larkin et al.
Representations also mentioned
Glaser emphasises the role of knowledge;
Robert Glaser, Education and Thinking: the role of
knowledge, American Psychologist, 39(2) 1984 93-104
“… the possession and utilization of an organized body of
conceptual and procedural knowledge, and a major component of
thinking is seen to be the possession of accessible and usable
knowledge.”
(cf. Marzano)
“… knowledge of novices is organised around the literal objects
explicitly given in a problem statement. Experts’ knowledge, on the
other hand, is organized around principles and abstractions that
subsume these objects.”
Glaser on representations
“We define a problem representation as a cognitive
structure … that is constructed by a solver on the basis of
domain-related knowledge and its organization. At the
initial stage of problem analysis the problem solver attempts
to ‘understand ‘ the problem by constructing an initial
problem representation”.
The use of representations constitutes part of the thinking!
Representations
A recurring, and perhaps dominant, theme in expert-novice
literature.
George M Bodner and Daniel S Domin,
Mental Models: The role of representations in problem
solving in chemistry
University Chemistry Education, 4(1) 2000 p24-30
Stieff, for example, reports that experts use a number of
strategies to solve organic chemistry problems. These
include:
[M Stieff and and Sonali Raje, Expertise & Spatial
Reasoning in Advanced Scientific problem Solving, Proc
8th Internantional conference on the Learning Sciences,
Utrecht, NL, 2008, p366-373]
initial visualization to understand the problem
visualizing
reasoning through images (generating new structures
through the mental use of)
as well as a host of analytical strategies.
M Suwa and B Tversky:
“External representations contribute to the
dynamic construction of ideas”
M Hegarty, B Meyer, H Narayanan, Diagrammatic
Representation and Inference, Proc. Inference 2002
Lecture Note ins Artificial Intelligence series,
Springer p341-343
Free working memory
Cue retrieval from long term memory
Allow perceptual judgements about spatial relations
Allow the generation of new ideas
Nersessian has integrated these ideas into a theory of
MODEL BASED REASONING
in which the construction and manipulation of iconic,
analogical mental models is considered as a form of
reasoning in its own right.
(Nancy J Nersessian, Creating Scientific Concepts, Bradford
books 2008)
Internal – external representations form a coupled
system which facilitates model-based reasoning!
Example, Stieff and Raje (2008):
10 experts, 7 tasks each
Take out Basic Recall and we are left with visualization and diagrammatic
reasoning as the principal mechanisms of problem solving.
Summary
Types of knowledge needed to solve problems:
 Declarative knowledge; domain specific content, facts,
laws, principles
 Procedural knowledge; production rules/sequences, how to.
 Conceptual knowledge; knowing why, understanding,
schemata
 Strategic knowledge; knowing when, where; heuristics
In addition
 Need experience in constructing, interpreting, manipulating
representations for analogical or diagrammatic reasoning
Two Questions:
Can we teach content knowledge
adequately?
Yes, but not easily
Can we teach problem-solving?
No definitive answer:
probably, but definitely
not as an adjunct to other
activities;
has to be integrated.
Field Dependence/Independence
Summarised by Witkin et al
Field-Dependent and Field-Independent Cognitive Styles
and Their Educational Implications Author(s): H. A. Witkin, C.
A. Moore, D. R. Goodenough, P. W. Cox Source: Review of
Educational Research, Vol. 47, No. 1 (Winter, 1977), pp. 1-64
FD most commonly characterised as an inability to
disembed information from its context (field)
Also involves an inability to see order in, or to superimpose
order on, a field – FDs takes their cues from others.
Continuum: extremes possible, should be viewed as
tendencies
Implications for problem-solving
Well documented that FD students struggle;
 cannot disembed the problem from its context
 working memory saturated from irrelevant
perceptual material
Deeper implication: students might not acquire sufficient
content knowledge.
Witkin discusses some educational implications, but two
examples from science will illustrate the point
Strawitz
Barbara M Strawitz, Cognitive Style and the acquisition and transfer of
the ability to control variables, J. Res. Sci. Teach. 21(2) 1984 133-141,
Barbara M Strawitz, Cognitive Style and the effects of two instructional
treatments on the acquisition and transfer of the ability to control
variables: a longitudinal study, J. Res. Sci. Teach. 21(8) 1984 833-841].
Teaching young teenagers the principles of scientific
experimentation, ie. controlling variables in a well defined
experimental task.
FI students pick up these skills.
FD students unable to develop these skills unless
instructed in an appropriate way.
With the right approach FD students are just as effective as
FI students
Antonietti and Gioletta
Allessandro Antonietta and Maria Alfonsa Gioletta, Individual
differences in analogical problem solving, Person. individ. Diff. 18(5)
1985 611-619,
Students were given either text-only or text-and-visual
instructional material which they had to use to solve a
problem.
FI students:
“The same diagrams were finally synthesized into one model”
“It was then easier to identify the existing relationships and isolate
relevant information despite its complexity”
FD students:
“Stated that too many cause-and-effect relationships were depicted in
the diagrams and described in the text, making it difficult to extract
what was relevant and solve the problem”
Recently FDI has been redefined as ability
rather than cognitive style.
Ability is distinct from general intelligence.
Field dependence–independence as
visuospatial and executive functioning in
working memory: implications for
instructional systems design and research
Kent A. Rittschof, Education Tech Research Dev
DOI 10.1007/s11423-008-9093-6
Conclusions
Expert – novice studies have elucidated
the nature of problem-solving within
specific domains; the role of recall, representations,
analogical reasoning
Marzano’s taxonomy allows an objective
to be associated with each task
Content knowledge and problem-solving
should be taught in an integrated manner
Illustration
Raluca Teodorescu, Cornelius Bennhold, and Jerry
Feldman, of George Washington University, have
constructed a taxonomy of introductory physics
problems (TIPP) based on Marzano’s taxonomy.
This is used to teach and assess problem solving in a
reformed physics class at GW
The following, taken from Jerry Feldmann’s
presentation at GIREP-EPEC 2009, summarises their
problem-solving strategy
Thanks to Jerry Feldman and Raluca Teodorescu, GW University
Fin.
Thank you
for your time.