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Research on Reasoning
Much fruitful research on reasoning has been generated
by the contradistinction between general reasoning skills
and knowledge of specific subject matter as instructional
goals. Some investigators have concentrated on identify-
ing the substantive knowledge needed for problem solving
in a particular area, others on the strategies used by
good problem solvers, and still others on impediments to
the use of reasoning skills.
An understanding of how a particular reasoning skill
operates in a specific substantive context can lead to
improved ability to teach the skill. For example,
Larkin and Reif (1976) analyzed what information a
learner should acquire while reading a description of a
scientific relation and used their findings to design
special training. In this training, students reading
scientific text practiced finding particular kinds of
information, e.g., units and symbols for quantities,
typical values, scaling properties, and features dis-
tinguishing them from other quantities. After they had
practiced answering such questions for a variety of
passages over a period of six weeks in a physics course,
the number of students who were able to learn how to use
a scientific relation from its written descriptions
increased from 40 to 80 percent. As another example,
Anderson (1981) has developed a computer model that
solves problems in geometry. Anderson's system is based
on a model of how experts solve geometry problems. This
model also can ~understand" a student's proposed solution
and give flexible advice when the student executes steps
that are either wrong or unproductive. Because the model
itself has an understanding of high school geometry, it
does not rigidly constrain the learner to particular
solution paths, but gives advice only when the student
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does something entirely incorrect or unproductive
Although it has not yet been proven, Anderson estimates
that an instructional system based on this model may
improve the efficiency of teaching geometry by a factor
Of two.
Knowledge required to solve arithmetic word problems
has been studied in detail.
.
Being skilled at such prob-
tems appears to be associated with being skilled not
merely in the basic arithmetical operations, but also in
categor iz ing problems according to the relationships they
involve. Riley et al. (1983) have categorized word prob-
lems according to the implicit problem structure. In
"change problems," a quantity (Jane's three marbles) is
increased or decreased (by Tom giving Jane two more or
taking two away). In "equalizing problems,. two quan-
tities must be considered and made the same (three of
Jane's seven marbles have to be taken away and added to
Tom's one marble). In "combine problems, n two separate
quantities have to be considered in combination (Jane's
marbles and Tom's marbles). In "compare problems, n
again, the quantities remain the same but have to be
compared (how many more marbles than Tom does Jane have?).
To solve such problems, students must recognize distinct
patterns that involve the ways in which quantities are
related. This research has been used to provide explana-
tions of the different levels of problem-solving skill
that are observed among elementary school children.
Schoenfeld (1979) has developed a theory of problem-
solving competence in mathematics that describes the
"executive" knowledge good problem solvers use to make
efficient use of their resources. The theory also deals
with attitudes of students about problem-solving tech-
niques that prevent them from using methods they have
mastered. Schoenfeld presented general heuristic
strategies (shown in Figure 1) that could facilitate
problem solving in mathematics to a group of college
science and mathematics majors and a control group. He
concluded that the likelihood of students' picking up
such strategies from experience is small and that
problem-solving strategies must be taught explicitly as
are other mathematical techniques. He also found that,
even when students master problem-solving techniques,
there is no guarantee that they will use them. Although
experts find these strategies easy to use, students must
be taught not only how to use them, but also when. When
students do use them, the impact on their problem solving
is substantial. However, much more research is required
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(1) Draw a diagram if at all Possible.
Even if you finally solve the problem by algebraic or other
means, a diagram can help give you a "feel. for the problem. It
may suggest ideas or plausible answers. You may even solve a
problem graphically.
(2) If there is an integer parameter, look for an inductive
argument.
Is there an "na or other parameter in the problem that takes
on integer values? If you need to find a formula for f(n), you
might try one of these:
(A)
(B)
(3)
Calculate f(l), f(2), f(3), f(4), f(5); list them in
order, and see if there's a pattern. If there is, you
might verify it by induction.
See what happens as you pass from n objects to n + 1. If
you can tell how to pass from f(n) to f(n + 1), you may
build up f(n) inductively.
Consider arguing by contradiction or contrapositive.
Contrapositive: instead of proving the statement waif X is
true then Y is true,. you can prove the equivalent statement waif
Y is false then X must be false.U
Contradiction: assume, for the sake of argument, that the
statement you would like to prove is false. Using this
assumption, go on to prove either that one of the given
conditions in the problem is false, that something you know to
be true is false, or that what you wish to prove is true. If
you can do any of these, you have proved what you want.
Both of these techniques are especially useful when you find
it difficult to begin a direct argument because you have little
to work with. If negating a statement gives you something solid
to manipulate, this may be the technique to use.
(4) Consider a similar problem with fewer variables.
If the problem has a large number of variables and is too
confusing to deal with comfortably, construct and solve a
similar problem with fewer variables. You may then be able to
tA)
Adapt the method of solution to the more complex problem.
(B) Take the result of the simpler problem and build up from
there.
(I)
Try to establish subgoals.
Can you obtain part of the answer, and perhaps go on from
there? Can you decompose the problem so that a number of easier
results can be combined to give you the total result you want?
FIGURE 1 The five problem-solving strategies.
Source: Schoenfeld (1979)
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to identify truly useful strategies, how experts learn
how and when to apply them, and efficient means of passing
this knowledge on to students.
A nether of studies have identified obstacles to
learning reasoning that lie in the preconceptions students
bring to school. For example, Champagne et al. (1980),
Clement (1982), and McCloskey (in press) have shown that
college students have preconceived notions about common
physical situations that run counter to and can interfere
with learning the principles of physics taught in the
classroom. Even after instruction, naive pre-Newtonian
beliefs about basic mechanics interfere with students'
understanding of the physical world. Various strong
convictions, which are often reinforced by unaided common-
sense perception (as in naive accounts of forces acting
on the movement of objects on an inclined plane), become
intertwined with new learning and inhibit its progress.
Alternative learning experiences can connect these
tendencies. By presenting computer simulations of events
(for example, objects falling in accordance both with
Newtonian principles and commonsense perceptions),
exper imental instruction in this area has been shown to
influence strongly held beliefs and deepen understanding
of the power of scientific accounts (Champagne et al. ,
1980; diSessa, 1982; White, 1984). Such results are
examples only, but they portend a deeper understanding of
the mechanisms underlying reasoning skills in science,
mathematics, and technology, and the development of
effective instructional strategies to teach these skills.
DEVELOPING COMPETENCE
To understand the mechanisms of reasoning skill, it
has proven useful to compare and contrast the performance
of more and less skilled individuals. Much more work
needs to be done, but good examples of such research
include that of Chase and Simon (1973), Egan and Schwartz
(1979), Chi et al. (1981), Jeffries et al. (1981), Clement
(1982), Lesgold (1983), and Voss et al. (1983). These
studies are concerned both with how a reasoning skill
operates and with the difficulties and limitations of
students who have not acquired this skill. The studies
indicate that problem solving proceeds on the basis of
the solver's representation of the problem. Students
with less skill tend to represent problems through
recognition of literal surface features and not by
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inferences from abstracted principles in the domain of
knowledge pertinent to a problem. Yet investigations in
radiology, architecture, electronics, chess playing, and
physics show that experts categorize problems according
to principles--this is a Newton 's-second-law problem, a
conservation-of-energy problem, etc.--rather than
according to the specific set of factors and conditions
presented by the problem. The relations between a
structured body of knowledge about the pertinent domain
and the problem-solving process is mediated through the
quality of the solver's representation. Both the scope
and depth of the solvers domain-related knowledge and
its organization--its completeness and its coherence--
determine the efficiency of progress toward the solution.
In addition, experts' knowledge includes information
about conditions of applicability for various procedures.
Average performers in a field often know enough about the
domain to construct an efficient initial representation
of a problem, but they lack knowledge of the conditions
for applying certain procedures.
A major research challenge is to understand better
the dynamic process through which reasoning skills are
acquired, to develop a rich theory of learning particu-
larly targeted toward reasoning skills that are difficult
to acquire. Such a theory would facilitate the develop-
ment of more effective interventions to help learners
acquire these skills more efficiently and reliably.
Productive work in this area includes that of Greeno
(1980), Anderson (1981), Larkin et al. (1983), and Riley
et al. (1983). Anderson's instructional system for
geometry, mentioned earlier, not only includes sufficient
knowledge to solve a wide range of geometry problems, but
it can also adjust the nature of this knowledge to match
the state of the learner. For example, for a beginner,
the system might break up the recognition of congruent
angles into several substeps. For a more advanced
learner, the system might expect the learner to recognize
congruent angles in a single step. Larkin has noted that
experts and novices solving textbook physics problems use
a very different set of subgoals. A typical novice,
however, exhibits a strategy that shows a few expert
features as well as a predominantly novice structure.
Thus it seems that learners gradually acquire the system
of subgoals that experts use productively. As another
example, Riley et al. have developed a model of how
children's knowledge for solving simple word problems
changes over time. Primitive versions of the model are
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based on extremely simple mental structures, while more
powerful versions can take advantage of more elaborate
structures.
In studies contrasting more and less skilled indi-
viduals, the more skilled individuals are often much
easier to understand. They have well-formed processes
and models of knowledge that correspond accurately to the
discipline. In contrast, less skilled individuals often
have processes and organizations of knowledge that are
rich and complex, unstable, and do not correspond
accurately to the discipline. Current research (McCloskey
et al., 1980; Clement, 1982; McCloskey and Kohl, 1982)
suggests that learners do not simply discard these
processes when they are instructed. Instead, their
initial reasoning processes interact with instruction in
complex ways. Research efforts should be targeted toward
understanding those initial processes and showing how
they can be addressed through instruction (Glaser, 1984).
One approach might be through confrontation so that new,
more effective processes can replace old. Another
approach might be through incorporation, where useful
aspects of initial processes can be incorporated into
more accurate and effective processes.
Cognitive studies use the notion of prototypical
knowledge structures or schemata to account for various
phenomena in memory, comprehension, problem solving, and
understanding. Schema theory attempts to describe how
acquired knowledge is organized and represented and how
such cognitive structures facilitate the use of knowledge
in particular ways. This theoretical construct has
particular utility for devising approaches to instruction
as individuals attempt to interpret new information on
the basis of prior knowledge. Modes of instruction that
demand interrogation of the learner's knowledge and
thinking and that demand confrontation with new knowledge
are being investigated by a number of researchers. For
example, Collins and Stevens (1982) have studied effective
teachers' procedures for teaching students domain-specific
rules and theories. The procedures involve shaping a line
of inquiry that helps students articulate their naive
initial conception of facts and principles and then
accept, modify, or reject them in the light of their
predictive power, congruence with new facts, and the like.
This inquiry approach both enables the student to assimi-
late new information efficiently and provides practice in
deriving rules or theories for related knowledge. An
important feature of the approach is the selection of
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cases and questions that enable students to use extant
knowledge as a framework for new learning.
Research toward better understanding of children's
developing capabilities for scientific reasoning is
needed, including more detail on the capabilities of
children at varying ages and on the kinds of scientific
experiences that aid the development of those capabil-
ities. Work on understanding scientific reasoning in
young children has illustrated that they are capable of
using quite intricate thought processes (Carey, 1985).
The nature of mathematical concepts on which young
children build their learning is beginning to be des-
cribed. Pervasive changes in children's reasoning and
learning abilities appear as they gain knowledge in
various domains. Researchers need to study what children
can do as well as what they cannot do. As Gelman and
Gallistel (1978:242) note: "the discoveries we have made
about development would not have been possible if we had
followed the trend of considering preschoolers merely as
beings who lacked the capacity of their older siblings.
Our hypothesis of more capacity than meets the eye has
served us well. We expect that researchers who keep
their eyes open will find still more unexpected ability
in young children.
The committee recommends research on how competence
in reasoning skills is acquired, including the mechanisms
of reasoning skills, particularly as evidenced in the
differences between novice and experienced learners; the
dynamic processes through which reasoning skills are
acquired in the context of specific domains of knowledge;
and the scientific reasoning skills of children.
THE SEARCH EOR GE~I~ IN CONING SKILLS
Despite many efforts to understand and teach
"generals reasoning skills, success has been unclear.
For example, Polya (1957) outlined some general help for
solving problems. He divided problem solving into four
phases--understanding, planning, executing, and looking
back--and formulated heuristics such as thinking of a
simpler problem that is similar to the difficult problem
at hand. While almost everyone finds these formulations
appealing, success in improving problem solving behavior
based on such general advice remains to be demonstrated.
Early computer models of problem solving were also quite
general . Ernst And Newell (1969), for example, developed
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a general problem solver" with computer codes for solving
problems that could be applied to a variety of specific
task domains, including algebra, geometry, several
puzzles, and logic. Again the example is appealing, but
this line of work has not developed into truly powerful
general problem solvers. One explanation may be that
effective reasoning requires a large amount of domain-
specific knowledge, as illustrated by the work cited in
the previous section, and relatively little general-
purpose knowledge. This conclusion is further supported
by considerable recent work on modeling and simulating
experts in various fields. The models generally do
require a large amount of domain-specific knowledge and
relatively little general-purpose knowledge (Davis et
al., 1977; Duda et al., 1978). Studies of problem
solving by cognitive scientists have indicated a strong
influence of specific knowledge structures on effective
performance in a discipline (Lesgold, 1983; Voss et al.,
1983).
Studies in the social sciences, particularly proto-
col analyses of expert and novice political scientists,
indicate that problem solving in these fields generally
proceeds through an analysis of the historical background
of the problem, the positing of a solution, and its con-
sideration in the light of the suburoblems it would oen-
erate. For example, experts who were asked to consider
the Soviet agricultural situation laid out the problem
first either in terms of its actual history or in terms
of long-standing ideological factors influencing agri-
cultural policy. Their solutions consisted of lines of
intersecting argumentation on subgroblems such as tech-
_
-
nological capacity, transportation, supply and demand,
and agricultural education. In this instance, experts
were differentiated from novices not only by the range of
subproblems they uncovered in examining the implications
of their solutions, but also in the quality and depth of
the knowledge they incorporated into their lines of
argument (doss et al., 19531.
The latter is important in
the social sciences, in which the test of a solution is
the strength of the argument since hypotheses can seldom
be experimentally validated.
It appears that problem solving and comprehension
are based on knowledge, and that people continually Try
to understand and think about the new in terms of what
they already know. If this is indeed the case, then it
seems best to teach reasoning skills--e.g., skills needed
for solving problems and for correcting errors of
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understanding--in terms of knowledge domains in which
individuals are attempting to become competent.
To further such teaching, the committee recommends
research on reasoning in particular disciplines, aimed at
understanding how abilities to make inferences, to reason,
and to generate new information can be fostered by ensur-
ing contact with prior knowledge that can be restructured
and further developed as learning takes place.
There is, however, research that suggests how par-
ticular general-purpose reasoning skills may operate
(Larkin and Reif, 1976; Schoenfeld, 1980; Larkin et al.,
1983; Palincsar and Brown, 1984). In these studies,
researchers have specified the skills to be taught in
detail and applied them to a focused set of related
domains. For example, in an integrated set of training
studies, Palincsar and Brown have shown that instruction
based on a system of inquiry can greatly increase
children's skills in understanding what they read--from
20 percent to 60 percent correct scores on reading
comprehension tests. The training sessions they designed
focus on encouraging children to internalize skills that
foster comprehension. These skills are modeled bythe
instructor, who leads dialogues involving paraphrasing
main ideas, questioning ambiguities, predicting questions
that are implicit in a given passage, and hypothesizing
the array of themes of a reading passage. The aim of the
sessions is to enable the children to lead such dialogues
and eventually develop techniques of self-criticism. An
important aspect of the instruction is keeping students
fully aware of the purpose of the activity, its utilities
for comprehension, how it improves their deficiencies,
how and when to use the various interrogative techniques,
and the expectation that they will eventually internalize
the techniques in their own performance.
Currently, significant effort is being devoted to
thinking and reasoning courses in secondary and post-
secondary education; the effectiveness of such instruc-
tion requires careful analysis and evaluation (Chipman et
al., in press; Segal et al., in press). Of particular
interest are the self-regulatory or metacognitive capabil-
ities present in mature learners. Examples of these
abilities include knowing what one does and does not
know, predicting the outcomes of one's performance,
planning ahead, efficiently apportioning time and cog-
nitive resources, and monitoring and tailoring one's
efforts to solve a problem or to learn (Brown, 1978).
These skills vary widely. Although individuals can be
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taught knowledge of subject-matter rules and procedures
or appropriate theories, if transfer of learning to new
situations is a criterion, then they need to know how to
monitor the use of this knowledge. Self-regulatory
activities thus become important candidates for instruc-
tion, and their presence may predict student abilities to
solve problems and to learn successfully.
Hence, the committee recommends focused research on
self-regulatory or metacognitive capabilities what they
are, how they develop, and how learners can be helped to
acquire them. We also recommend systematic tracking of
outcomes resulting from efforts to teach generalized
thinking and reasoning skills. . .
Representative terms from entire chapter:
skilled individuals
