Running head: INDUCTIVE REASONING TRAINING
Inductive Reasoning: A Training Approach
Karl Josef Klauer
Technical University of Aachen
Gary D. Phye
Iowa State University
Germany
Revised manuscript # 354
USA
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Abstract
For several decades, researchers have been engaged in examining inductive reasoning
in order to identify different cognitive processes participants use when they are dealing
with inductive problems. In many cases the processing observed is unique to the
problem encountered. The present article presents a different approach. A prescriptive
theory of inductive reasoning is presented that identifies cognitive processing which
basically consists of a procedural strategy for making comparisons (i. e., of looking for
similarity and difference). Based on the prescriptive theory of inductive reasoning, it is
hypothesized that training in the use of the procedural inductive reasoning strategy
will improve cognitive functioning in terms of, (a) increased fluid intelligence
performance, and (b) better academic learning of classroom subject matter. The
following review and meta-analysis summarizes the results of seventy-four training
experiments based upon nearly 3,600 children. Both hypotheses are confirmed.
Further, two moderating effects were observed, (a) training effects on intelligence test
performance increased over time, and (b) positive problem-solving transfer to
academic learning is greater than transfer to intelligence test performance. Moreover it
is shown that the results cannot be explained by placebo or test coaching effects. It can
be concluded that the proposed strategy is theoretically as well as educationally
promising and that children of a broad range of age and of intellectual capacity benefit
when participating in such training.
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Inductive Reasoning: A Training Approach
Empirical research in inductive reasoning began about a hundred years ago in the
context of intelligence research when Spearman found that his g factor of general intelligence
was mainly determined by inductive processes “eduction of relations”, (Spearman, 1923).
Later, dimension analytic research also identified inductive processes as central intellectual
factors identified as Reasoning (Thurstone, 1938), or Fluid Intelligence (Cattell, 1963). Using
modern linear structural equations, Gustafsson (1984; Gustafsson & Undheim, 1992) came to
comparable conclusions.
Meanwhile, in psychology and education the research focus has evolved to the
analysis of cognitive processing when students solve inductive reasoning and other types of
problems. Many researchers in the cognitivistic tradition have been engaged in exploring
inductive processes (Goldman & Pellegrino, 1984; Pellegrino & Glaser, 1980; Sternberg &
Gardner, 1983). More specifically, research has focused on the cognitive processing involved
in series completion (Holzman, Pellegrino, & Glaser, 1983), analogies (Alexander & Willson,
1987; Gitomer, Curtis, Glaser, & Lensky, 1987; Pierce, Duncan, Gholson, Glen, & Kamhi,
1993), classifications, (Coley, Hayes, Lawson, Moloney, 2004; Tennyson, Youngers, &
Suebsonthi, 1983; van de Vijver, 2002), categorizations (Heit & Hayes, 2005; Sloutsky &
Fisher, 2004), and matrices (Carpenter, Just, & Shell, 1990).
Moreover, educational psychologists have studied ways of fostering inductive
reasoning and its transfer (Phye, 1989; 1990; 1997). Other studies have focused on the use of
special inductive measures in teaching and learning, such as analogies as instructional tools in
teaching science (Chen, 1999; Gentner, 1989; Gick & Holyoak, 1983; Robin & Mayer, 1993)
or other subject matter (Bean, Searles, Singer, & Cowen, 1990; Reed, Dempster, & Ettinger,
1985; Vosniadou & Schommer, 1988).
In addition, researchers in the field of artificial intelligence have constructed computer
programs based upon process models that aim to solve certain kinds of problems in order to
test their theories of inductive reasoning (Ernst & Newell, 1969; Holland, Holyoak, Nisbett,
& Thagard, 1986; Kotovsky & Simon, 1973). Even sophisticated mathematical models have
been developed and tested that are able to predict how people process inductive problems, for
instance causal models (Rehder, 2003; Rehder & Burnett, 2005) or Bayesian models (Heit,
2000). Recent cognitive process research has highlighted the impact of general principles that
seem to strongly influence subjects’ inductive processes (Heit & Feeney, 2005; Heit & Hahn,
2001; Medin & Heit, 1999). This line of research shares some important features with the
research to be reported here by stressing principles such as similarities and diversities.
In contrast to most of the research mentioned, the following prescriptive theory does
not claim to analyze how learners proceed cognitively when they solve inductive problems.
For example, which processes are activated, or in what way processes are modified by
properties of the given problems. This line of research is particularly demanding because
processes employed can vary depending on the participants and their special experiences as
well as the influence of different kinds of problems. Our focus is comparably modest in that
the prescriptive theory delineates a rather simple strategy that in principal should enable
subjects to solve any inductive problem. And it is rather easily tested by teaching participants
to make use of the recommended strategy.
A Prescriptive Theory of Inductive Reasoning
At the outset it is useful to distinguish between inductive reasoning and inductive inferring.
Inductive reasoning is aimed at detecting generalizations, rules, or regularities. For example, if
a number of objects is given and if it is found that all of these are toys made of wood, a
generalization or regularity has been discovered. Should we extend this generalization to the
totality of toys by stating that all toys are made of wood, then we would have made an inductive
inference, although a false one in this case. An inductive inference extends the generalization
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beyond the scope of experience by asserting something about a non-observed or even nonobservable universe of objects. Drawing inductive inferences is much more demanding but also
much more critical than inductive reasoning that might precede it.
The main purpose of this article is to describe a recent prescriptive theory of inductive
reasoning (not inductive inferring) and to test this theory for its usefulness in educational
research, teaching, and training. The theory was developed some years ago (Klauer, 1989;
1991; 1993; Klauer & Phye, 1994; Klauer, Willmes, & Phye, 2002) and since a number of
experiments have been performed by different authors, it seems reasonable to review and
empirically evaluate the basic assumptions.
The first step of the theory is to define inductive reasoning. A useful formulation has
been provided by Glaser and Pellegrino (1982, p. 200) who stated, “All inductive reasoning
tasks have the same basic form or generic property requiring that the individual induce a rule
governing a set of elements.” There is general agreement that tasks such as, (a)
classifications, (b) analogies, (c) incomplete series, and (d) matrices require inductive
reasoning and that they are widely accepted as typical inductive reasoning tasks (Büchel &
Scharnhorst, 1993). It is commonly accepted that these four types of tasks require the
detection of a rule or, more generally, of a regularity. However, is this list of the four types of
tasks an exhaustive one? Is there a plausible reason why only these four tasks are identified
as inductive reasoning tasks? In addition, is inductive reasoning characterized by individual
instances of (a) its product, (b) the detection of a rule, or (c) characterized by a certain kind of
process? Or, is it defined by some combination of the three dimensions? Figure 1 suggests
some answers and in some respects, a more specified definition (Klauer, 1989; Klauer &
Phye, 1994).
Inductive reasoning consists of detecting regularities and irregularities by finding out
A
B
a1 similarity
b1 attributes
a2 difference
of
a3 similarity &
b2 relations
difference
C
c1 verbal
c2 pictorial
with
c3 geometrical
c4 numerical
c5 other
Figure 1. Definition of inductive reasoning.
material.
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5
According to Figure 1, inductive reasoning not only reveals regularities but also
irregularities and diversities. For instance, in cases where a rule only partially governs a set
of elements, the assumed rule has to be rejected and, possibly, replaced by a better fitting one.
Moreover, using three facets A, B, and C, Figure 1 specifies by which means a rule can be
detected or rejected, namely by a comparison process. Comparing is defined as finding out
similarities and differences, or both (Tversky, 1977). Hence, facet A is the comparison facet.
The comparison process produces regularities consisting of at least one commonality among
all the objects. According to facet B that commonality refers either to attributes of the objects
or to relations between objects. We call facet B the category facet. Looking at modern logic,
another aspect can be introduced since attributes are identified as predicates with one
argument, while relations are identified as predicates with two or more arguments. Since no
other predicates are possible, the distinction implies that attributes and relations exhaust all
possibilities for characterizing objects. This fact demonstrates the far reaching impact of
inductive reasoning. In Figure 1, facet C of the definition is the materials facet. Facet C
specifies the nature of the inductive reasoning materials. It is, of course, possible to replace
the categories of facet C with school relevant material such as types of subject matter taught
in school.
The central facets of the definition are facets A and B. They clearly constitute six
classes of inductive reasoning, not considering all possible combinations. The six classes are
specified in Table 1, where item formats are given as they are identified in current intelligence
tests. The first three classes are varieties of classification tasks, while the remaining can be
identified as analogies, series, and matrices. Thus, it becomes clear that the traditional item
format possibilities reflect all inductive reasoning tasks. It is evident from Figure 1 why this is
the case.
Table 1 specifies the names attributed to the six classes, the facet identifications, the
item formats as found on intelligence tests, and the cognitive processes required to solve these
items. The relationships among the six basic varieties of inductive reasoning tasks are
depicted in the genealogy of Figure 2.
CROSS
CLASSIFICATION
GENERALIZATION
Similarity
SYSTEM
CONSTRUCTION
DISCRIMINATION RECOGNIZING
DIFFERENTIATING
RELATIONSHIPS
RELATIONSHIPS
Difference
Similarity
Attributes
Difference
Relationships
STRATEGY OF INDUCTIVE REASONING
Figure 2. Genealogy of tasks in inductive reasoning.
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Table 1. Types of Inductive Reasoning Problems
Process
Generalization
Facet
Item
Cognitive Operation
Identification
Formats
Required
class formation
similarity of
class expansion
attributes
a1b1
finding common
attributes
Discrimination
Cross
a2b1
a3b1
Classification
Recognizing
a1b2
Relationships
identifying
discrimination of
disturbing
attributes (concept
items
differentiation)
4-fold scheme
similarity &
6-fold scheme
difference in
9-fold scheme
attributes
series completion
similarity of
ordered series
relationships
analogy
Differentiating
a2b2
disturbed series
Relationships
System
Construction
differences in
relationships
a3b2
matrices
similarity &
difference in
relationships
Depending on the problem given, the strategy to reason inductively requires a person
to scrutinize either attributes of the objects or the relations among them. Hence, Figure 2
shows two branches, which are divided again into two branches depending if one is looking
for similarities or for differences. In some cases, both similarities and differences are called
for, bringing the two branches together again. A symmetrical figure results because the
attribute and the relations branches are similarly differentiated.
From the definition portrayed in Figure 1, it should be possible to design an analytic
strategy that enables one to solve every kind of inductive reasoning problem. Its basic core
would be a comparison procedure. The objects (or, in case of relationships, the pairs, triples
etc. of objects) would be checked systematically, predicate by predicate (attribute by attribute
or relation by relation), in order to find out commonalities and/or diversities. Presumably, a
computer program could be developed to solve any problem of inductive reasoning.
Inductive Reasoning
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However, human beings may prefer to make use of a heuristic strategy, as depicted in Figure
3. In this case, a participant starts with a more global inspection of the task and builds a
hypothesis. This can then be tested so that the solution might be found more rapidly,
depending of the quality of the hypothesis. In a training program, participants might be
advised to first use the heuristic strategy and, if some attempts do not lead to a solution, to
then apply the analytic strategy.
Start
Compare the objects
(pairs of objects) globally
Try again
Build a hypothesis
Test the hypothesis
by directed comparisons
Rule discovered?
yes
End
no
Already tried
several times ?
no
yes
Apply the analytical
strategy
Figure 3. Heuristic or hypothesis – guided strategy of inductive reasoning.
Hence, Klauer’s theory of inductive reasoning first offers a definition of inductive
reasoning. This definition leads to an exhaustive classification of inductive reasoning tasks.
Moreover, it specifies processes by which these task types can be solved. Finally, the
cognitive process analysis leads to two comprehensive strategies that problem solvers might
use when solving inductive problems.
However, as was mentioned earlier, it is not claimed that all learners always proceed
according to the analytic or the heuristic strategy. Actually, one can assume that people make
use of innumerable ways of solving different varieties of inductive tasks. What follows from
the definition of Figure 1 concerning the solving process is not a description of what
commonly occurs but a prescription of how to proceed in order to effectively and efficiently
solve inductive problems (i.e., the theory is basically a prescriptive one). Consequently, an
adequate test of this kind of theory is to teach participants to apply it and to see whether they
are able to solve inductive problems more adequately than those that have not had the
opportunity to learn how to proceed. Thus, training experiments are appropriate means for
testing the theory.
Training Programs
According to the theory introduced, training to reason inductively provides an opportunity for
participants to acquire the basic strategy of inductive reasoning, to modify it appropriately for
the six varieties of inductive tasks and to experience sufficient opportunities through practice
Inductive Reasoning
to internalize the strategy. Actually, participants should be able to recognize an inductive
problem whenever they meet one. More precisely, they do not have to be able to
Table 2. Instructional Objectives of the Ten Lessons
Lesson
1
Training objective
Solving the problems naively
Comment
The children should get familiar with
the material and the training situation
2
Distinguishing attributes and
relationships
Introduction of the terms “attribute”
and “relationship.” Sorting all items
of lesson 1 appropriately
3
Recognizing the three attribute
classes
Distinguishing the three classes.
Sorting all of the attribute items thus
far
4
Recognizing the three relationship
classes
Distinguishing the three classes.
Sorting all of the relationship items
thus far. Recapitulation of the
attribute problems
5
Solving and checking procedures
with similarity problems
Learning how to solve and check
generalization and recognizing
relationship problems. Recapitulating
sorting of items
6
Solving and checking procedures
with difference problems
Learning how to solve and check
discrimination and differentiation of
relationship problems. Recapitulating
sorting of items
7
Solving and checking procedures
with similarity & difference
problems
Learning how to solve and check
cross-classification and system
construction problems. Recapitulating
sorting of items
8
Repeating and practicing problems
of the attribution branch
Rehearsing all of the processes with
attribution problems
9
Repeating and practicing problems
of the relation branch
Rehearsing all of the processes with
relation problems
10
Mixed repetition of all kinds of
problems and procedures
Practicing all types of identifying,
solving, and checking processes with
all types of problems
classify a given inductive problem as belonging to one of the six varieties. It is enough when
they are able to identify a problem as one similar to a familiar problem and then assign the
adequate solving strategy. Ideally, the application of the inductive strategy should be
8
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automatized so that whenever the problem solver comes across an inductive problem, he or
she automatically chooses the adequate strategy.
Three training programs and their corresponding manuals have been developed,
• Program I for children of the ages 5 – 8,
• Program II for children aged 11 – 13, and
• Program III for youth of the ages 14 – 16
(Klauer, 1989; 1991; 1993). Program I is basically nonverbal so that it can be used in all
language environments. For this reason an American version of the manual has been
published (Klauer & Phye, 1994) as well as a Dutch version (Klauer, Resing, & Slenders,
1996). The first two programs are appropriately used with regular classroom children, gifted
children and learning disabled children. The cognitive development of the child would
determine whether Programs I or II were to be used for training. Program III was designed to
be used with mildly learning disabled youth, with weak performances in school and who are
at risk for vocational integration.
Training Format
Each of the programs I, II, and III consist of 120 items, that is, 20 items for each of the
6 basic classes of inductive reasoning tasks. Further, programs II and III offer forty verbal,
forty figural, and forty numerical problems adapted to children’s everyday life experiences
and to the problems they might meet in school. Only program I differs a bit from this scheme
since it is not anticipated that these children are able to read. Also, at the beginning of
program I a few problems with real blocks are included so that the children can manipulate
the blocks when solving the problem at hand.
According to recommendations by Belmont, Butterfield and Ferretti (1982), during
training children should receive ample opportunities to acquire the appropriate metacognitive
aspects of the solving procedure which are mentioned in Table 2. As a rule, a complete
training episode is made up of 10 lessons with 12 items each. With programs II and III,
trainers are advised to adopt the plan outlined in Table 2 which specifies objectives for each
of the ten lessons. Beginning with lesson two, metacognitive aspects are at the center of
attention. In lessons two and three, problem classes are defined by attributes while in lesson
four problem classes are defined by relationships. Children learn the terms for attributes and
relationships and they are provided the opportunity to classify all of the training problem
tasks they have encountered. Lessons five to seven repeat what has been learned so far but in
a different order. This way the children are provided the opportunity to realize that problems
can differ with respect to the category involved (attributes or relations) but can require
identical processes (looking for similarity or for difference or for both). The last three lessons
provide review and practice to help students to consolidate what they have learned.
Various kinds of verbal self-regulating instructions are helpful and it is useful to give
participants tips and hints such as, “COMPARING means looking for SIMILARITIES and
DIFFERENCE". During the last three lessons, children are encouraged to acquire a habit of
monitoring themselves and their solution processes. Three procedural processing questions
identified below should be asked and students should be expected to answer appropriately to
all three queries for each new problem.
QUESTION
1) What do I have to look at?
2) What should I do to find the solution?
3) How can I check my solution?
ANSWER
Similarity or difference or both
with attributes or relationships.
Compare, (i. e. look on similarity
Or difference or both). I do it
according to an assumption or
systematically.
By the opposite comparison.*
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*The opposite comparison is a favored checking procedure. When, for instance, all objects
are characterized by a certain attribute, no difference must be found with respect to that
attribute.
Meta-analysis
Currently, 74 experiments (Appendix A) have been performed where at least one
training group participated in a training of inductive reasoning using one of the three training
programs and where at least one other group did not receive such a training but continued
regular classes or another activity. Under these circumstances it is advisable to use metaanalysis in order to gain an overview of the most important results. Based upon the
prescriptive theory of inductive reasoning and a review of the research literature cited, it is
possible to derive certain hypotheses which can be tested meta-analytically using the available
data base.
Hypotheses
As already mentioned, a strong research tradition has shown that inductive reasoning
is a central part of general intelligence. Snow, Kyllonen, and Marshalek (1984) were able to
clearly demonstrate this fact. They used the data set of Thurstone (1938) and reanalyzed it via
multidimensional scaling. This reanalysis found inductive and deductive reasoning to make
up the core of fluid ability. Consequently, today tests of fluid intelligence contain at least
some inductive subtests [e g Cattell Culture Fair Tests (Cattell & Cattell, 1963) or Raven
Progressive Matrices (Raven, Court, & Raven, 1994)]. It is worth noting however that the
inductive training programs and these intelligence tests are quite different in terms of test
items. The training programs offer meaningful material and incorporate problems that
children may encounter in their daily lives. In contrast, the intelligence tests consist of
abstract, isolated and more or less meaningless material. Our first and central hypothesis deals
with the effect of the training on intelligence test performance. It is necessary for theoretical
reasons to test whether the training improves intellectual functioning. However, because
intelligence has a positive impact on learning in school, academic learning is also of interest
from a practical point of view.
Hypothesis 1. It is expected that inductive reasoning training results in positive
transfer to tests which measure fluid intelligence (effectiveness hypothesis). According to this
hypothesis a positive transfer effect of the training program to a standardized adequate g
factor intelligence test can be viewed as evidence of the effectiveness of inductive reasoning
training. As is usually the case in meta-analyses, possible moderator variables will also be
addressed. A case in point is in reference to the comparison of the effectiveness of the three
programs. If the three programs are valid constructs of the same theoretical conception, then
one expects their effects do not differ substantially from one another. However, the results
concerning the three programs are not independent of several potential moderator variables.
Primary examples are the age or level of cognitive development of the children being
trained which may exert special influence. Program I is used in kindergarten, school
kindergarten and primary school. School kindergarten is a special German institution. It is
designed for children old enough to enter school but who are not yet ready for regular
schooling. Thus, they are generally fostered for another year in a kindergarten–like
environment in the school but without exposure to the three R’s. For instance, these mildly
learning disabled or otherwise exceptional children may be less responsive to a cognitive
training than the learning disabled but older children in special education settings in the
primary school. Program II is used in secondary schools whereas Program III is used with
even older but mildly learning impaired youths. Thus, Program III results are confounded not
only with age but also with a slightly reduced level of general ability. One would expect that
both the chronological age and the differing levels of cognitive development of the students
across programs may account for differences in mean effect sizes. This is particularly the case
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with respect to possible aptitude by treatment interactions (Goska & Ackerman, 1996). Thus,
it can be assumed that participant differences play a role in moderating the effect of the
inductive training on intelligence test performance. Despite confounded variables and with
respect to possible future research, it makes sense to check whether age and/or participant
differences modify the effectiveness of the training.
The social condition of the training may also play a moderating role. One can assume
that a one–to–one training is the most effective because the contact between trainer and
trainee is most intense and because the trainer can adapt his or her interventions optimally to
the child’s individual needs. However, one-to-one teaching has not always been found to be
the most effective approach with every subject matter or skill (Elbaum, Vaughn, Hughes &
Moody, 2000). The training of pairs of children might be both effective and efficient and is
advantageous for several reasons (Lochhead, 1985). For example, if one of the children is
asked to solve a given problem and comments loudly on his or her attempts, then the other
partner can be asked to check whether all of the relevant information has been correctly
considered. Thus, both children are cooperatively involved in the solving process, but each
has different roles. According to Lochhead’s recommendation, children can change their
mutual roles with the next problem. This way a child learns to apply techniques and
strategies he or she did not previously know and both acquire metacognitive vocabulary and a
habit of reflecting metacognitively on their own inductive reasoning processes. Lochhead’s
principles can also be applied when one deals with small groups of trainees (Palincsar &
Brown, 1984) or even with whole classes. For practical purposes it would be advantageous if
small groups or intact classes could be trained simultaneously and effectively.
Finally, a fourth moderator variable should be taken into consideration, the authorship
of the experiments. Nearly half of the training studies are published by Klauer. These
experiments were conducted by Klauer’s staff members or by his students of psychology or
education in fulfilling their requirements for a master’s degree. While none of Klauer’s
students received different instruction other than that which is available to anyone else
reading the handbook, one cannot rule out the assumption that the experiments published by
Klauer show differing effect sizes. All of the other experiments were conducted and published
by different persons and hence are not subject to the same criticism.
Hypothesis 2. The effects of inductive training on intelligence test performance equals
an effect due merely to participation in training irrespective of what is trained (placebo
hypothesis). Actually, it has been suggested that the positive results of inductive training may
result from a variation of the placebo effect (Hager & Hasselhorn, 1995; Hager, Hübner, &
Hasselhorn, 2000). These authors assume that a close bonding between trainer and trainee is
developed if a single child or a small group of children is trained. This special relationship
and the individual attention children experience during training may account for the cognitive
training effects, irrespective of the kind of cognitive activity the children are experiencing.
According to this assumption, the decisive agent is not the special training but the close
personal relationship between trainer and trainee which results when children participate in
any training. Hence, it is necessary to determine whether the positive performance effects of
inductive training can be attributed to non-inductive reasoning factors encountered during
training.
Hypothesis 3. The effect of inductive reasoning training on intelligence test
performance does not disappear after a few weeks (durability hypothesis). As Lipsey and
Wilson (1993) demonstrated meta-analytically, placebo effects (if observed) disappear after a
few weeks. Given our stance with reference to hypothesis 2, we expect a much longer lasting
effect of inductive training. Otherwise such training would be a waste of children’s time if it
does not lead to a lasting improvement in cognitive functioning. This is especially the case
with respect to positive transfer of training to academic learning. A rapidly diminishing effect
of the training would not be of great value to educational practice. Thus, for theoretical as
Inductive Reasoning
12
well as practical reasons, an important question is whether training effects will last for some
time or whether they will rapidly disappear.
Hypothesis 4. Training in inductive reasoning will result in positive transfer to the
learning of academic subject matter (transfer hypothesis). However, the transfer effect will be
smaller than the effect of the training on intelligence test performance. One can assume that
inductive reasoning training improves learning to the extent it improves intellectual
functioning and information processing. Moreover, because nearly all regular classroom
subject matter requires the acquisition of generalizations (be it in the form of concepts,
classes, rules or laws), one could anticipate that inductive reasoning training would improve
learning of many kinds of subject matter. Such a line of reasoning is encouraged by Csapó’s
(1997) cross–sectional research, where he found a close relationship between inductive
reasoning and science learning in school. However, because intelligence test performance
correlates with scholastic achievement only to a moderate degree, the transfer effect on
learning of regular subject matter in school is anticipated to be smaller than the effect of the
training of intelligence test performance.
The four moderator variables that potentially modify intelligence test performance
(hypothesis 1) can also be checked for impact on positive transfer to academic learning. It is
possible that across the three programs, the age or cognitive development of the children, the
social condition of the training, or the group of authors might also moderate the transfer effect
on academic learning (hypothesis 4).
Hypothesis 5. The training effects are not due to procedures of coaching or teaching to
the test (coaching hypothesis). Hager, Hübner, and Hasselhorn (2000) have suggested the
possibility that the positive training effects can be the result of teaching to the test or
coaching procedures. Coaching to a test is an old and widespread practice, particularly where
test performance determines admission to certain careers (Anastasi, 1981). However,
participants in Klauer’s training programs are instructed to solve six item classes, which are
found in many intelligence tests. During training participants are not taught only to solve
specific items but a general strategy “compare and contrast,” which has to be adapted to the
six classes of inductive items.
Hasselhorn (1995) recommended distinguishing between coaching or training effects
according to the nature of the effects. According to Hasselhorn, coaching results only in
improvement of performance, while training results in improvement of the underlying
competence. He proposes two indicators of improvement in competence, namely transfer to
other variables and durability of the effects. Hence, the coaching hypothesis will be tested
within the context of two preceding hypotheses, the durability and the transfer hypothesis.
The Meta-Analysis Data Pool
Klauer’s theory of inductive reasoning and the training programs have attracted the
attention of many researchers. Actually, the theory of inductive reasoning and the first
training experiments caused some controversies and discussions. As a result, by the end of
2004, a total of 74 experiments were available which used one of the three previously
described training programs (references can be found in Appendix A). Note that in some of
the 71 published articles, more than one experiment is reported. Unfortunately, only a few of
the articles were published in English language journals.
The probability is rather high that all European evaluations published are included in
the analysis because the community of interested researchers is well known. In order to
identify additional relevant research, systematic internet searches were performed, mainly
using the database PsychINFO and labels such as “Inductive reasoning AND training,”
“Klauer AND training,” and “Denktraining,” the German label of the programs. These
searches produced 180 hits. However, none of these searches led to the identification of even
one additional paper not already included in the data pool.
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13
Only three of the experiments in the data pool (see Appendix A) were not published,
the studies by Brünken (# 56), Bussas (#7), and Favre (#55). These experiments were theses
and were not published mainly because they did not add new insights or results to the body of
research which was available at that time.
Method
For a study to be included in the meta-analyses testing hypotheses 1-5, one of two
kinds of experiments had to have been employed. Studies either had to be: (a) a test of the
positive transfer of one of Klauer’s inductive reasoning training programs to intelligence test
performance, or (b) studies had to be a test of the positive transfer of one of Klauer’s training
programs to learning of academic subject matter.
Training Study Designs
Designs to test hypotheses 1-3 focus on transfer to intelligence test performance. The
central hypothesis (1), predicts transfer of inductive reasoning training to performance on
intelligence tests which include measures of fluid intelligence. All of the studies included in
this review employed a two-group design. Frequently, this took the form of a training group
contrasted with a no–training control group. These comparison groups continued with regular
kindergarten work or schooling (i.e., they received academic training, but not the specific
inductive training). In these studies, the effect of the special training was compared with the
effect of regular schooling.
Sometimes the two-group design involved two treatment groups. In these cases a
group trained inductively was contrasted to a group trained with a non-inductive program, (i.
e., with an alternative training program). Thus, the effect of the inductive reasoning training
program is compared with the effect of a different training program. Here both treatment
groups share the experience of participating in a training program. However, such a two
training groups design without a no-training control group sometimes turns out to be
disadvantageous. When there is no difference between both treatments, one cannot decide if
both treatments were equally effective or equally ineffective. Fortunately, a number of studies
made use of a three-group design to assess the effect of the treatments by adding a regular notraining control group to the two training groups. If both treatments yield more or less the
same effects, comparisons with the no–treatment control group enables one to decide whether
both treatments are equally effective or ineffective.
Furthermore, the three-group design enables one to assess the amount of effect that
may be attributable specifically to the training situation. Hypothesis 2, the placebo hypothesis,
states that the experience of participating in any training can, in and of itself, produce positive
effects on intelligence test performance. As a test of hypothesis 2, different kinds of
alternative non-inductive training programs were used, [training of spatial cognition with
program Tetris (Masendorf, 1994; Souvignier, 1997), training of metacognitive strategies
using problem-solving with non-inductive problems (Bornemann, 1989; Klauer, 1992;
Kolmsee, 1989), arithmetical training (Angerhoefer et al., 1992), training in reading strategies
(Klauer, 1996), training in other academic learning strategies, social games (Sonntag, 2004),
and a motivational training (Fries, Lund & Rheinberg, 1999)].
Training Episode
Most of the training studies lasted for several weeks. In a typical training episode, two
lessons are given per week, requiring a training period of five weeks. As a rule, subjects were
tested a week before training and a week after training (pretest – posttest design), resulting in
a total training study duration of seven weeks. A few of the training studies needed less time,
while others needed more time, depending on the given circumstances.
All of the training experiments used such a pretest–posttest design. Additionally, in
order to test hypothesis 3 concerning the durability of the training effects, after the posttest
was administered a second follow-up posttest was administered some weeks or even several
months later. Very often, although not in every case, pretest, posttest and follow-up tests
Inductive Reasoning
14
were identical. This design element leads to considerable retest effects. However, in cases
where a no–training control group or an alternatively trained control group was available for
comparative purposes, specific training effects are not confounded with retest effects.
Following training, intelligence tests were administered that consisted entirely of
inductive reasoning items such as Raven Coloured Progressive Matrices (CPM), Standard
Progressive Matrices (SPM), or Advanced Progressive Matrices (APM). Others included
subtests of inductive reasoning items , Cattell Culture Fair Tests (CFT), the German version
of the Cognitive Ability Test (CAT), or the German version of the Columbia Mental Maturity
Test (CMMT) for learning disabled children. In some studies only those subtests which
require inductive reasoning were administered whereas others included the complete test.
Thus, the data upon which the meta-analysis is based refer to intelligence tests which in part,
or in their entirety, measure inductive reasoning. Further, the overwhelming majority of the
intelligence tests made use of abstract, more or less meaningless material so that the stimulus
tasks for training (real world ) and transfer (abstract materials) were quite different in terms of
similarity and familiarity.
Designs to test hypothesis 4 that focuses on the transfer to academic learning. Of the
74 training experiments available, 38 studies tested the effect of the training on learning of an
academic subject matter. These studies consisted of two phases. During the first phase,
children in a class were randomly assigned to one of two groups, a training group and a notraining control group. Most training groups participated in inductive training for two sessions
a week over a five week period while the control groups continued regular classes during
these sessions.
In the second phase of these studies, a lesson followed involving a subject matter that
belonged to the regular curriculum that had not yet been taught. In participating classrooms,
these lessons were administered to both the training and control group together. This second
phase was experienced as regular classroom activity by both experimental group and control
group students. As a rule, after choosing an appropriate subject matter, an informal criterionreferenced test was developed and used as both pretest before and as posttest after the
common lesson. This way it was possible to measure how much the children already knew
and how much they acquired during the lesson. Specifically, it was possible to determine
whether the inductive training and the no-training control groups differed in academic
learning.
Across studies, a variety of academic topics were chosen: mathematics, biology,
geography, physics, as well as reading, spelling, grammar, and learning of foreign languages
or learning and problem solving in ecology. In Appendix B, the disciplines are listed in
column DV 2 (dependent variable 2) as numbers 1 - 7. The numbers 8 – 13 deal with
cognitive variables, which do not include learning of a subject matter. The verbal descriptors
for variables coded 1 – 13 in Appendix B are provided in the description of the table.
Typically, data were analyzed using an analysis of covariance with the pretest as
covariate, training as the independent variable, and the posttest as the dependent variable.
Some authors preferred, however, t-tests to ascertain significance of posttest performance and,
following this, significance of differential pretest-posttest increase for the training and control
groups.
Research Design Shortcomings
The overwhelming majority of the training experiments included only small samples
of children or youth because it is more efficient to conduct the training of inductive reasoning
in small groups rather than the entire class. This is a trade-off. While training can be more
effectively administered to small groups, statistically significant findings may be less likely to
result. Fortunately, the information provided by effect sizes will help determine if there is a
coincidence of small N and small effect size that inevitably leads to insignificant results.
Actually, across studies one would not expect all of the trainers to be equally effective.
Inductive Reasoning
15
Consequently, if many small groups are trained by many trainers who differ in effectiveness
then a relatively large range of results is anticipated.
In a few cases whole classes were assigned to the treatments. Here we are dealing
with quasi-experiments even when the classes were randomly assigned to the conditions.
However, because pretest-posttest designs were used it was possible to take the most
important pre-experimental differences into consideration.
Calculating Effect Sizes
There are differing opinions on whether it makes sense to calculate an effect size when
the effect is not statistically significant (Cahan, 2000; Levin & Robinson, 2000; Robinson &
Levin, 1997; Wainer & Robinson, 2003). It is clear that with small sample sizes only large
effects can reach the level of statistical significance. Regardless, for these meta–analyses it
seems meaningful to estimate all of the effect sizes. Furthermore, because of their theoretical
and practical importance, in the following analyses, we added to each mean effect size its
confidence interval (Thompson, 2002).
Due to differences among research designs, different effect size estimates were
generated and are discussed below. Since the early efforts of Cohen (1968) and Glass (1976),
effect sizes have typically been estimated as standardized mean differences. Also, following
suggestions by Hedges and Olkin (1985) the difference of the means is divided by the pooled
standard deviation of both groups.
Calculating g. For all g-measure comparisons (Hedges & Olkin, 1985), g was
calculated as:
MT – M C
g = ─────
sp
Because we are often dealing with small sample sizes and because pretest means can vary
considerably in this case, even when the subjects were randomly assigned to the treatments,
we decided to improve the effect size estimations by correcting for pretest differences. Thus,
gcorr is calculated for each effect size estimation.
gcorr = gposttest – gpretest
The three g columns of the Appendix B contain gcorr values with the following
headings: g11 refers to the effect size on the first dependent variable (an intelligence test) as
posttest; g12 refers to the same variable as follow–up some months later; and g2 represents the
transfer effect on a second dependent variable (often school relevant learning).
Calculating d. Hedges (1981), however, showed that g overestimates the true effect
size a bit if one deals with small samples. Therefore d is used instead of g when further
calculations were computed,
3
d = (1- ───────) g
(4N – 2) - 1
where N refers to the number of subjects involved. Primarily dependent on sample size N, d
is somewhat smaller than g. Moreover, when mean effect sizes are estimated, this is done
using the weighted integration method by Hedges and Olkin (1985, p.112). This means the d
values are weighted according to their sample sizes so that studies with larger samples get a
higher weight than studies with smaller samples. This procedure generally leads to lower
mean effect sizes. Hedges and Olkin also developed correcting procedures in order to account
for a lack of test reliability. Since applying these corrective procedures would lead to higher
means, corrections for attenuation were not employed.
Inductive Reasoning
16
When Effect Sizes are Dependent. If one compares the results of one experimental
group with the results of one control group per experiment, no dependency exists among the
contrasts. However, in our case we have 74 experiments but 89 contrasts as can be seen in
Appendix B or in Table 3 below. In order to avoid overlap as much as possible, one should
differentiate among three types of contrasts. In Appendix B, the three types are depicted in
column “Contrast” and denoted by 1, 2, or 3. Here are the explanations.
Contrast type 1: An inductively trained group is contrasted with a no – training control group.
Contrast type 2: An inductively trained group is contrasted with an alternatively trained
group.
Contrast type 3: An alternatively trained (non-inductive reasoning training) group is
contrasted with a no – training control group.
Contrast 1 is the primary contrast for providing credible results. For contrast 2 an
inductive training group is tested against a different but non-inductive training group.
Contrast 3 enables one to directly test hypothesis 2, the placebo effect. Arguably, if this
contrast yields significant training results it is merely the result of participation in any training
of any type that accounts for the observed improvement in cognitive performance. This would
suggest that the special attention trainees receive during training is the agent of change.
Table 3
Frequency of Contrasts Used in the Meta-Analysis
(In parentheses experiment reference numbers according to Appendix B)
________________________________________________________________________
Type of
Frequency of
Frequency of
contrast
experiments
effect sizes
Comment
________________________________________________________________________
1 only
49
49
nonproblematic*
2 only
13
13
nonproblematic
1+3
6 (39, 33, 23, 8, 5, 3)
12
nonproblematic
1+1
3 (64, 34, 22)
6
problematic**
1+2+3
1 (73)
3
problematic
1+1+3
2 (52, 9)
6
problematic
Sum
74
89
* Nonproblematic: The estimates of the effect sizes are independent of each other.
** Problematic: The estimates of the effect sizes are not independent of each other.
With respect to dependency, no problems arise when only one contrast per experiment
is tested, be it contrast 1, 2, or 3. When two dependent variables were employed in a single
experiment, for instance an intelligence test and a criterion-referenced subject matter test,
both contrasts were never included in the same meta-analysis because they referred to
different hypotheses. So, no dependency problem occurred in these cases. The same holds
true when contrasts 1 and 3 were made. Contrast 3 deals with a different research question.
However, when both contrasts 1 and 2 were performed in one experiment, there is a
problem of dependency because the same control group is used twice. The same is true when
two type 1 contrasts are calculated (e. g., when two different varieties of inductive training
were compared to a control group that did not receive training). Again, in these cases, the
effect size estimations refer to the same control group and, hence, are not independent from
each other. Yet, such research is of practical importance because it helps provide evidence
concerning the effectiveness of particular types of inductive training programs. To eliminate
these results from analysis would be shortsighted from an educational perspective. Because
only six experiments and 15 effect size estimations were affected, it was decided to keep
them, while including only half the number of subjects in the control groups. This procedure
Inductive Reasoning
17
Frequency of Effects
has the following ramifications: (a) the d values as well as the corresponding means receive
lower estimations, (b) d values are corrected for (larger) sampling errors, and (c) the means
are based upon correct instead of inflated sample sizes so that their confidence intervals are
not overestimated.
Results
According to hypothesis 1 it was predicted that inductive training would result in
positive transfer effects to tests measuring fluid intelligence. The results of the 74 experiments
are summarized in Appendix B. Omitting experiment number seventy because no intelligence
test was administered, 73 experiments are available where the effect of an inductive training
on intelligence test performance can be ascertained. Based upon these experiments, 79
contrasts could be performed using an intelligence test as the dependent variable. The effect
sizes g11 ranged from -0.05 to 1.30 with an unweighted mean of Mg = 0.59 ± 0.31 (n = 79, N =
3595). Figure 4 complements these data by presenting a visual representation of the
frequencies for the various effect sizes observed and the shape of the distribution of
frequencies.
Figure 4. Distribution of the g effect sizes of training transferring to intelligence test
performance.
Figure 4 shows a rather symmetrical distribution. As expected, there is considerable
variability among the effect sizes. Some are quite small, while others are rather large.
However, the bulk of the data clusters about the mean. Nevertheless, the effect sizes are not
normally distributed about their mean (p = 0.048, Kolmogorov - Smirnov test with Lilliefors'
correction). There were too many relatively large effect sizes. If one eliminates the three
largest effect sizes as possible outliers, the effect sizes are normally distributed. Regardless,
the following analyses are based upon the whole data set.
Hedges and Olkin (1985) have shown that effect measure g slightly overestimates the
actual effect sizes, particularly when dealing with rather small samples. Effect measure d
gives an unbiased and typically a somewhat lower estimation of the effect size. Using the
above mentioned weighted method by Hedges and Olkin, several means d+ were calculated
which weigh the single d values according to the respective sample sizes N. Table 4 provides
an overview of the results with respect to the effect of the training of inductive reasoning on
intelligence test performance. These results are disaggregated in terms of various moderator
variable influences.
According to Table 4, the overall weighted mean of the 79 effect sizes d+ equals 0.52.
As expected, this value is a bit smaller than the above mentioned unweighted mean g (Mg =
0.59). Nevertheless, one can conclude that on average an inductive training improves
Inductive Reasoning
18
intelligence test performance by about half a standard deviation. This corresponds to an
improvement of about 20 percentile ranks for the average participant. Moreover, all of the
average d+ effect sizes in Table 4 differ significantly from zero (p < 0.01). The same is true
for the average d+ values of Table 5.
Also, the coefficient Q of homogeneity was calculated and its probability ascertained.
If this probability lies beyond a significance level, then the effect sizes, combined to produce
a common estimation, are heterogeneous. If this is the case, it makes sense to look for
variables moderating the effects. As shown in Table 4, all of the mean effect sizes are
accepted as being based on a homogenous body of data. This suggests that the considerable
variability observed can, possibly, be explained by sampling errors. In any case, there is no
requirement to analyze further the influence of moderator variables.
Nevertheless it is not unusual in meta-analyses to refer to anticipated possible
moderator variables. In our case the three Klauer programs are of interest as were the type of
students involved, the training conditions, and the two varieties of authorship. None of these
moderators taken as a whole seem to have a particular impact on the results. The only
disaggregated exception involves the nine studies that were performed in kindergarten that
show a tendency to produce heterogeneous results within the group.
Table 4. Weighted Means d+ of the Effects of the Inductive Training on Intelligence Test
Performance: Summary of Meta – Analyses
(n: Number of Contrasts, N: Number of Subjects)
_____________________________________________________________________
Variable
d+ 95 % Confidence Interval
n
N
p (Q)*
______________________________________________________________________
All experiments
0.52
0.46 – 0.59
79
3 595
0.80
______________________________________________________________________
Possible Moderator Variables
Programs
Program 1
0.57
0.49 – 0.66
42 2 004
0.65
Program 2
0.43
0.32 – 0.55
24 1 144
0.63
Program 3
0.50
0.31 – 0.67
13
447
0.99
79 3 595
Subjects
Kindergarten
0.47
0.24 – 0.70
9
306
0.09
School kindergarten 0.43
0.11 – 0.75
5
153
0.95
Primary school
0.61
0.50 – 0.72
19 1 274
0.63
Secondary school
0.42
0.30 – 0.54
24 1 148
0.78
Special education
0.54
0.39 – 0.69
22
714
0.98
79 3 595
Training Conditions
One-to-one
0.34
0.15 – 0.53
11
432
0.70
Pairs
0.59
0.41 – 0.78
15
466
0.64
Small groups
0.57
0.46 – 0.67
36 1 422
0.84
Classes
0.51
0.40 – 0.62
17 1 275
0.50
79 3 595
Authorship
Staff Klauer
0.57
0.47 – 0.68
35 1 386
0.79
Other authors
0.49
0.40 - 0.57
44 2 209
0.80
79 3 595
_______________________________________________________________________
* Probability of the coefficient Q of homogeneity
Inductive Reasoning
19
Hypothesis 2 (placebo effect) assumes that the effect of inductive training is due
simply to the special social conditions of any training. In order to test this hypothesis nine
experiments were planned as three group designs. The first group received the regular
inductive training, the second group was alternatively trained (but not inductive reasoning),
and the third group did not participate in a training at all but continued regular classes. As was
reported in the method section, the alternative training sessions made use of a broad range of
different materials and activities so that the tests are not restricted to a single set of conditions.
Hypothesis 2 is tested by comparing the alternatively trained group with the no–training
control group (i. e., by a type 3 contrast). A meta–analysis on these nine effect sizes leads to a
mean effect size estimation of d+ = 0.004 (n = 9, N = 230) which is not significantly different
from zero. Hypothesis 2 is clearly rejected. One can conclude that the non–inductive training
procedures had no significant effect on intelligence test performance.
Testing Hypothesis 3 is an attempt to determine how long training effects on
intelligence test performance will last. What about its durability? In order to address this
question, 22 experiments involving 1094 students were administered a follow-up posttest. As
a rule, the first posttest was administered a few days after the end of the training. Follow–up
posttests were administered between 3 and 15 months later (see column Month in Appendix
B). The correlation between (a) the follow-up posttest and (b) the months between first
posttest and follow-up posttest when (c) the pretest values were partialed out is r12.3 = 0.44 (df
= 19), p = 0.045). Unexpectedly, this result means that training effects do not diminish over
time and even increase slightly when the dependent variable is performance on fluid
intelligence test items.
Academic Learning
As previously mentioned, in 38 experiments inductive training was followed by a
lesson on a new subject matter that was a part of the regular curriculum. Training and control
groups participated in the same lesson, and the amount of learning could be ascertained by
criterion-referenced pre- and posttests. Effect sizes are displayed in the last column (g2) of the
table in Appendix B. The following analyses refer to this column. The mean effect on
academic learning was Mg = 0.74 ± 0.36 (n = 38, N = 1723) and thus larger than the mean
effect on intelligence (Mg = 0.59 ± 0.31). In Figure 5 the distribution of the effect sizes on
academic learning are depicted. It conveys the impression that there are a few outliers with
unusually large effect sizes. Nevertheless, the analyses were performed with the whole data
set.
Under these circumstances it again makes sense to look for possible moderator
variables. This was done using the more adequate d measure of effect size and its means
calculated with the weighted method of Hedges and Olkin (1985). The results of the meta–
analysis used to examine hypothesis 4 are summarized in Table 5.
20
Frequency of Effects
Inductive Reasoning
Figure 5. Distribution of the g effect sizes of training transferring to academic learning
performance.
Table 5 shows an overall mean effect size d+ = 0.69. This is significantly larger than
the corresponding mean effect d+ = 0.52 for the transfer of training effects on intelligence test
performance (p < 0.05, which is in contract to expectations. Obviously, inductive reasoning
training improves academic learning of school–type subject matter more than it improves
measured intellectual functioning. Moreover, with p(Q) = 0.35, one can maintain the
hypothesis that the whole body of effect sizes is homogeneous and that it is not necessary to
look further for moderator variable effects. However, upon closer inspection, one moderator
variable effect requires additional consideration. Within program III, eleven experiments did
not produce homogeneous effect sizes (p(Q) = 0.04). The largest outlier effect sizes on
learning were found in studies with older students in special education who were trained using
program III.
Table 5. Weighted Means d+ of the Effects of the Inductive Training on Learning: Summary
of Meta – Analyses
(n: Number of Contrasts, N: Number of Subjects)
_______________________________________________________________
Variable
d+ 95 % Confidence Interval
n
N
p (Q)*
______________________________________________________________________
All experiments
0.69
0.59 – 0.79
38
1 723
0.35
______________________________________________________________________
Possible Moderator Variables
Programs
Program 1
0.64
0.49 - 0.80
11
663
0.89
Program 2
0.64
0.49 – 0.79
16
698
0.87
Program 3
0.84
0.62 – 1.06
11
362
0.04
38
1 723
Subjects
Primary school
0.63
0.47 – 0.80
8
594
0.71
Secondary school 0.59
0.43 – 0.75
15
650
0.90
Special education 0.94
0.74 – 1.14
13
434
0.20
36
1 678
Training Conditions
Inductive Reasoning
Small groups
Classes
0.73
0.62
0.60 – 0.86
0.46 – 0.79
Authorship
Staff Klauer
Other authors
0.67
0.71
0.54 – 0.81
0.57 – 0.86
22
11
33
969
612
1 581
21
0.38
0.76
20
966
0.79
18
757
0.80
38
1 723
_______________________________________________________________________
* Probability of the coefficient Q of homogeneity
Prior to analyzing the social conditions of the training, five experiments are excluded
from the analyses because of too few subjects (three studies with single children and two with
pairs of children) so that 33 experiments were subjected to analysis. Results indicate that
training in small groups of three to five children turns out to be a bit more effective, with
training of whole classes close behind. In terms of effectiveness and practical utility, this is a
noteworthy result. Finally, it is of some interest to again look at the authorship of the
publications. This time the other authors slightly outperformed Klauer’s staff members
although the difference is negligible.
Discussion
With hypothesis 1 it was predicted that training in inductive reasoning would result in
positive transfer performance to tests of fluid intelligence. Results support this hypothesis.
Looking at Figure 4, one can assume that training in inductive reasoning does no harm and
often benefits children’s intellectual development.
According to hypothesis 2, it is assumed that the effects of the training are brought
about by merely participating in any training activity, irrespective of what activities are
trained (placebo effect). The nine experiments in which this issue was specifically addressed
leads to the conclusion that unspecific placebo effects do not play a role. Clearly, hypothesis 2
can be rejected. This result is in line with placebo studies which have been reported in
educational settings (Adair, Sharpe, & Huynh, 1990) and other contexts (Dush, Hirt, &
Schroeder, 1989). Moreover, as Lipsey and Wilson (1993) demonstrated meta-analytically,
when placebo effects are observed they disappear rather quickly. All in all, one can conclude
that the results reported so far cannot be explained by placebo effects.
Another question pertains to the assumption that the effects of training diminish over
time (see hypothesis 3 concerning durability of the effects). Actually, present results suggest
that the effects in some of the reviewed studies are stable or increase linearly over time. How
should such results be explained? One possibility would be that the control children’s
intellectual capacities decrease linearly in time. However, there is no reason for such an
assumption. Another possibility would be that the trained children make more and more use
of the acquired strategy as a result of its successful employment and we are observing
instances of self-regulated learning. Support for the durability of effects hypothesis would be
strengthened if future research in this area were designed to include no-treatment control
groups. In the set of studies reviewed, no information is available whether the effect might
change beyond the time span of 15 months.
Incidentally, in the European research literature not included in the current review and
meta-analyses, some authors have assumed long lasting and cumulative effects of similar
interventions that have been deemed “snowball effects” (Feuerstein, Rand, Hoffman, &
Miller, 1980). Acquiring a general strategy is said to foster future learning, which in turn
should improve even later learning so that the gap between trained and untrained participants
could get larger and larger. The alleged mechanisms leading to such results are termed
“causes of other effects as well” (Clarke & Clarke, 1989; Schweinhart & Weikart, 1980, p.
64).We have not accepted such a position.
Inductive Reasoning
22
As to hypothesis 4, transfer effects on academic learning, the pattern of results are
very encouraging. The effects are unexpectedly high and typically larger than the effects on
intelligence. The transfer effects were observed over a broad range of academic subject matter
One reason for this success could be that the strategy taught during training can be directly
applied in many academic situations. For example, each academic subject requires concept
formation and every concept is defined through a set of common attributes. Moreover, every
academic subject matter contains rules, laws, or regularities which are defined through one or
more common relations. Hence, one can assume that the inductive reasoning strategy as it is
acquired during training can be applied and transferred to classroom learning performance.
Actually, the prescriptive strategy is a rather simple procedure that can easily be
learned. At its center is basically a general procedure of comparing, (i.e., looking for
commonalities and diversities). During the ten lessons, teaching deals exclusively with the
concepts required, the cognitive and metacognitive processes involved, and the application of
this knowledge to new problems in a wide variety of situations. However, these
considerations would not necessarily predict that inductive reasoning training fosters learning
of a subject matter to the same or to an even larger extent than it improves fluid intelligence
test performance. This question should be addressed by new research projects.
As to the teaching to the test or coaching hypothesis (hypothesis 5), it was tested by
two criteria, (a) the durability of the effects, and (b) the transfer to academic learning tasks.
Both criteria are clearly fulfilled so that the interpretation of positive training results as being
due solely to test coaching procedures can be rejected.
A look back at the theory of inductive reasoning presented
In summary, it seems appropriate to have a look back on the theory underlying this
research. The core of the theory is prescriptive in nature. It states that inductive reasoning can
be achieved by a comparison strategy, where attributes of objects or relations between objects
are to be scanned with respect to similarity, difference, or both, for commonality and/or
diversity. It is not claimed that subjects actually proceed this way when they solve inductive
problems. Instead, our contention is only that subjects have a good chance to solve inductive
problems more effectively when they make use of the comparing strategy. The results show
that this is the case. Moreover, it was found that the comparing strategy not only transfers to
intelligence test performance but that it improves intellectual competencies. Furthermore, it
also improves problem-solving and learning of academic subject matter.
The theory, however, maintains that the comparing strategy enables one to solve all
kinds and varieties of inductive reasoning problems. But the empirical data show that the
trained participants by no means are able to solve all types of inductive problems to which
they are exposed. To explain the gap between actual and theoretical improvement one could
assume that the comparing strategy may be a helpful but not a sufficient condition for solving
all inductive reasoning problems. A simple example can demonstrate that the strategy is not
sufficient for every problem. If one has to find commonalities between a pan, a lemon, and a
microwave oven, then one needs to have some special knowledge, and a reasonable solution
is not possible if that knowledge is not available. Also, insufficient knowledge may not be the
only problem that can lead to failure in the application of the inductive strategy. Regardless of
participants’ cognitive ability level, for successful application of the comparative strategy it
must be mastered at a sufficiently high level of proficiency. Support for this contention is
provided by Klauer (1996) where it was shown that a number of “above average” children
who had not appropriately learned the comparing strategy benefited little from training. One
must conclude that a strategy which is effective in some cases will not lead to success for
everybody in all situations.
Finally, if the prescriptive theory of inductive reasoning works as expected, one can
assume that teachers should be able to apply it in their regular classes. The basic principle
“compare and contrast” is very concrete and teachers should be able to adapt it to their regular
Inductive Reasoning
23
lessons instead of relying on published training programs. This way they would have the
opportunity to adapt the strategy according to their student’s level of cognitive development
as well as to the subject matter being taught. The majority of the training studies were
conducted by educationally inexperienced graduate students of psychology or education and
one can assume that experienced teachers would be able to teach the comparing strategy and
its applications with much greater effectiveness.
Research, of an experimental or quasi-experimental nature, seeking to replicate these
inductive reasoning training effects in U.S. schools, is encouraged. In principle, one should
expect both an improvement of children’s intellectual competencies and more efficient
learning of regular subject matter as has been found in the experiments reviewed.
Inductive Reasoning
24
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Inductive Reasoning
Author Notes
Karl Josef Klauer is professor emeritus of education at the Institute of Education, Technical
University of Aachen, Germany. He is especially interested in the research of teaching and
learning. E-mail: [email protected]
Gary D. Phye is professor of curriculum/instruction and psychology in the College of Human
Sciences of Iowa State University, Ames, Iowa. He is especially interested in problem-solving
transfer and academic learning. E-Mail: [email protected]
Gary D Phye
N162b Lagomarcino
Iowa State University
Ames, IA. 50011-3190
Correspondence concerning this article may be addressed to either author at the aforementioned
E-Mail addresses.
30
Inductive Reasoning
31
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Inductive Reasoning
35
20.
Hager W., & Hasselhorn, M. (1993). Induktives Denken oder elementares
Wahrnehmen? [Inductive reasoning or fundamental perception?]. Empirische
Pädagogik, 7, 421-458.
19.
Beck, M., Lüttmann, B., & Rogalla, U. (1993). Wenn Du denkst, Du denkst... Eine
Untersuchung der Effektivität des Klauer´schen Denktraining [If you think you think
… A study on the effectiveness of Klauer’s Training to Reason]. Zeitschrift für
Enwicklungspsychologie und Pädagogische Psychologie, 25, 297-306.
18.
Hager, W., & Hasselhorn, M. (1993). Evaluation von Trainingsmaßnahmen am
Beispiel von Klauers Denktraining für Kinder [Evaluation of training measures with
the example of Klauer’s Training to Reason for Children I] . Zeitschrift für
Entwicklungspsychologie und Pädagogische Psychologie, 25, 307-321.
17.
Klauer, K. J. (1993). Über den Einfluss eines Trainings zum induktiven Denken auf
den Erwerb und die Nutzung der Lernstrategie des "Networking" [Impact of a training
of inductive reasoning on the acquisition and use of the learning strategy
„networking“]. Zeitschrift für Entwicklungspsychologie und Pädagogische
Psychologie, 25, 333-352.
16.
Klauer, K. J. (1993). Induktives Denken beeinflusst das Rechtschreiblernen [Inductive
reasoning influences learning to spell]. Zeitschrift für Entwicklungspsychologie und
Pädagogische Psychologie, 25, 352-365.
14.-15. Klauer, K. J. (1993). Denken und Lernen bei Lernbehinderten: Fördert das Training
des induktiven Denkens schulisches Lernen? [Thinking and learning with slightly
retarded children: Does a training of inductive reasoning improve learning in school?].
Heilpädagogische Forschung, 19, 55-66.
13.
Klauer, K. J. (1993). Über die Auswirkungen eines Trainings zum induktiven Denken
auf zentrale Komponenten der Fremdsprachenlernfähigkeit [On the effects of a
training of inductive reasoning on central components of the ability to learn foreign
languages]. Zeitschrift für Pädagogische Psychologie, 7, 1-9.
12.
Klauer, K. J. (1992). Teaching inductive thinking to highly able children. European
Journal for High Ability, 3, 164-180.
10.-11. Klauer, K. J. (1992). "Bottom up" oder "top down"? Über die Transferwirkungen
zweier Strategien zum Training des induktiven Denkens [„Bottom up“ or „top down“?
On the transfer effect of two strategies of a training to reason inductively]. Sprache &
Kognition, 11, 91-103.
9.
Angerhoefer, U., Kullik, U., & Masendorf, F. (1992). Denk- und Rechenförderung
lernbeeinträchtigter Kinder: Multivariate Änderungsbeurteilung mittels PrädiktionsKFA [Fostering thinking and math performance with learning impaired children:
Multivariate evaluation of change by prediction CFA]. Psychologie in Erziehung und
Unterricht, 39, 190-195.
8.
Klauer, K. J. (1992). Problemlösestrategien im experimentellen Vergleich: Effekte
einer allgemeinen und einer bereichsspezifischen Strategie [Comparing problem
Inductive Reasoning
36
solving strategies experimentally: Effects of a general and a domain specific strategy].
In H. Mandl, & H.F. Friedrich (Hrsg.), Lern- und Denkstrategien, (pp. 58-78).
Göttingen: Verlag für Psychologie.
7.
Bussas, K. (1992). Erprobung des Denktrainings II im Gymnasium [Testing Training
to Reason II in a grammar school]. Unpublished master’s thesis, University of Aachen,
Aachen, Germany.
6.
Igelmund, T. (1990). Erprobung des Denktrainings II im vierten Schuljahr [Testing
Training to Reason II in a fourth class]. In K. J. Klauer, Denktraining für Kinder 1 (p.
64). Göttingen: Verlag für Psychologie.
5.
Kolmsee, R. (1989). Erprobung des Denktrainings I im ersten Schuljahr [Testing
Training to Reason I in a first class]. In K. J. Klauer, Denktraining für Kinder 1 (pp.
64-65). Göttingen: Verlag für Psychologie.
4.
Johnen, M. (1989). Erprobung des Denktrainings im Kindergarten [Testing the
Training to Reason in kindergarten]. In K. J. Klauer, Denktraining für Kinder 1 (p.
75). Göttingen: Verlag für Psychologie.
3.
Bornemann, K. (1989). Erprobung des Denktrainings in Zweiergruppen [Testing the
Training to Reason with pairs of children]. In K. J. Klauer, Denktraining für Kinder 1
(p. 73-74). Göttingen: Verlag für Psychologie.
2.
Bornemann, K. (1988). Erprobung des Denktrainings im Kindergarten [Testing the
Training to Reason in kindergarten]. In K. J. Klauer, Denktraining für Kinder 1 (p.
73). Göttingen: Verlag für Psychologie.
1.
Bornemann, K. (1988). Erprobung des Denktrainings im Kindergarten [Testing the
Training to Reason in kindergarten]. In K. J. Klauer, Denktraining für Kinder 1 (pp.
72-73). Göttingen: Verlag für Psychologie.
Inductive Reasoning
37
Appendix B
Summary Table of Meta – Analyses
Row Exper. Author N Subj. Contrast Progr. Soc. Cond. DV 1
89
74
0
30
5
1
3
3
2
88
73
0
39
5
1
3
3
2
87
73
0
38
5
2
3
3
2
86
73
0
20
5
3
3
3
2
85
72
0
36
5
1
3
3
5
84
71
0
29
5
1
2
3
5
83
70
0
40
5
1
2
3
82
69
0
79
3
1
1a)
2
4
a)
81
68
0
31
5
1
1
2
4
80
67
1
166 3
1
1
3
4
79
66
0
54
3
1
1
3
2
78
65
0
62
3
1
1
3
2
77
64
0
91
3
1
1a)
4
1
a)
76
64
0
97
3
1
1
4
1
75
63
0
156 4
1
2b)
4
2c)
74
62
0
32
3
2
1
1
2c)
73
61
0
18
1
2
1
1
2
72
60
1
48
4
1
2
4
2
69
59
0
24
5
1
3
3
5
68
58
0
24
5
1
1
1
5
71
57
1
46
4
1
2
4
2
70
56
1
25
2
2
1
2
4
67
55
0
31
4
1
3
3
5
66
54
0
30
4
1
3
3
5
65
53
0
40
5
1
3
3
2
64
52
0
51
4
1
2b)
4
2
63
52
0
29
4
1
2
4
2
62
52
0
39
4
3
2
4
2
61
51
0
24
1
1
1
2
3
60
50
1
54
3
2
1
3
4
59
49
1
40
4
2
1
2
7
58
48
0
47
3
1
1
4
4
57
47
0
40
5
1
2
3
5
56
46
1
41
3
2
1
2
2
55
45
0
331 3
1
1
4
1
54
44
0
39
3
1
1
4
5
53
43
0
28
4
1
2
3
2
52
42
0
32
5
1
1a)
2
8
51
41
0
40
4
1
2
3
2
50
40
1
48
4
1
2
3
6
49
39
0
29
5
1
3
4
5
48
39
0
29
5
3
3
4
5
47
38
1
28
3
1
1
3
4
46
37
0
43
3
1
1
3
2
45
36
1
60
4
1
2
4
3
44
35
1
84
4
2
2
4
6
43
34
1
22
3
1
1
3
2
42
34
1
22
3
1
1a)
3
2
41
33
1
22
4
3
2
3
5
40
33
1
22
4
1
2
3
5
39
32
0
68
4
1
2
3
2
g11 Months g12
DV 2
0.64*
6
0.80* 2
0.25
2
0.49
2
-0,22
2
0.49*
2
1.06*
2
2
0.24
11
0.22
11
0.90*
6
1.05* 2
0.83*
0.81*
5
0.34
2
0.69*
15
1.22* 4
0.59*
15
0.73* 4
0.27*
4
0.23
11
d)
0.05
13
0.33
8
0.35*
3
0.83*
2
0.47
2
0.39*
7
0.71*
12
0.73* 2
0.83*
3
0.88* 4
0.59*
3
0.59* 4
0.46*
0.50*
0.09
-0,14
1.24*
0.30
5
0.49*
7
0.54* 9
0.67*
4
0.50* 2
0.83*
1
0.46*
5
9
0.56*
1.00*
4
0.83*
0.42*
0.76*
11
0.60*
0.48*
4
0.45*
7
0.26
1.13*
9
1.51* 2
0.39*
4
0.54* 2
1.13*
1
0.31*
1
0.76*
0.59*
-0,17
6
-0,18
5
0.80*
6
1.09* 5
-0,05
10
g2
0.84*
1.68*
1.50*
-0,28
0.82*
1.33*
0.62*
0.16
1.31*
0.81*
0.77*
0.59*
0.69*
0.44*
0.51*d)
0.37
0.35
0.19
0.72*
0.53*
0.62*
0.45*
0.19
0.46*
1.24*
0.47*
1.07*
0.75*
0.70
0.53*
0.96*
0.80*
0.18
0.74*
0.42*
0.73*
0.48*
0.16
Inductive Reasoning
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
31
30
29
28
27
26
25
24
23
23
22
22
21
20
19
18
17
16
15
14
13
12
11
10
9
9
9
8
8
7
6
5
5
4
3
3
2
1
0
1
1
1
0
0
0
0
0
0
1
1
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
1
1
1
1
1
60
32
36
45
28
34
34
32
20
20
35
34
20
30
140
32
44
51
32
36
61
16
30
24
15
15
20
20
20
56
20
20
19
19
22
22
20
27
5
4
5
5
5
2
5
2
5
5
4
4
1
2
1
2
4
4
5
5
4
1
4
4
5
5
5
3
3
4
3
3
3
1
1
1
1
1
1
1
2
2
2
2
1
2
1
3
1
1
1
2
1
2
1
1
1
1
1
1
1
1
1
1e)
3
1
3
1
1
1
3
1
1
3
1
1
1
2
3
3
2
1
1
1
1
1
2a)
2a)
1
1
1
1
2
2
3
3
2
1
2
2
1
1
1
1
1
2
1
1
1
1
1
1
1
1
1
3
3
3
1
2
3
1
3
3
4
4
1
1
1
1
3
3
2
4
3
1
3
3
3
3
3
2
2
4
3
2
2
2
2
2
2
2
2
2
5
5
2
2
2
2
2
2
6
6
2
2
3
2
2
2
5
2
6
4
6
6
2
2
2
2
2
5
3
3
3
4
4
4
4
3
*) p ≤ 0.05
a)
b)
c)
d)
e)
Modified version of the program
Combination of the inductive and a motivational training
CFT subtests 3-5, i.e. the inductive subtests only
dpost instead of dcorr
Contrasted to a different control group
0.49*
0.23
0.32*
0.59*
0.61*
0.43
0.74*
0.28
1.29*
0.32
0.78*
0.38*
0.98*
0.51*
0.13
0.34
0.31
0.59*
0.76*
0.19
0.48*
1.24*
0.80*
0.15
0.24
0,58*
-0,45
1.00*
0.12
0.32*
0.43*
0.45
0.31
1.15*
1.30*
0.26
0.54*
1.12*
6
0.39
4
0.71*
3
5
10
4
7
7
0.89*
0.51*
0.34
0.99*
0.34*
0.39*
38
5
7
0.60*
1,37*
8
0.36
9
12
12
3
3
2
9
0.21
1.00*
1.01*
1.13*
0.56*
0.97*
0.04
3
5
2
2
6
1.00*
0.89*
0.35
1.11*
0.65*
2
0.56*
Inductive Reasoning
39
Legend for the columns of the summary table
Row
Number of contrast
Exper.
Number of the experiment in the Appendix List
Author
1: experiment was performed by students or staff members of Klauer
0: experiment was performed by other authors
Number of subjects
N
Subj.
Contrast
Progr.
Soc. Cond.
DV 1
g11
Months
g12
DV 2
g2
Kind of subjects (1: kindergarten, 2: school kindergarten, 3: primary school,
4: secondary school [incl. 9 experiments in grammar schools], 5: special
education)
Kind of contrast
1: inductive training vs. no training control group
2: inductive training vs. another but not inductive training
3: another training vs. no training control group
1: program I, 2: program II, 3: program III
Social condition of the training
1: One – to –one training, 2: Training of pairs of children, 3: Training in small
groups of 3-5 children, 4: Training of intact classes
First dependent variable
1: Informal Test of Inductive reasoning ITIR, 2: CFT by Cattell (German
version), 3: Cognitive Abilities Test CAT (German version), 4: Coloured
Progressive Matrices CPM, 5: Standard Progressive Matrices SPM, 6:
Advanced Progressive Matrices APM, 7: Columbia Mental Maturity Test
CMM for mildly retarded children (German version), 8: another intelligence
test.
Effect size g of dependent variable 1 immediately after training (corrected for
pre-experimental differences).
Time interval between g11 and g12.
Effect size g of dependent variable 1 the stated number of months after g11
(corrected for pre-experimental differences).
Second dependent variable
1: Learning and problem solving of ecology, 2: Learning and problem solving
of mathematics, 3: Learning and problem solving of biology, 4: Learning and
problem solving of geography, 5: Learning and problem solving in reading or
spelling or grammar, 6: Learning of foreign languages, 7: Learning and
problem solving of physics, 8: Frostig Test, 9: Memory test, 10: Vocational
Counseling Test (BBT 4-6), 11: Nonverbal and /or non-inductive intelligence
test, 12: Test of spatial reasoning, 13: Visual Discrimination Test POD.
Effect size g of the dependent variable 2 (corrected for pre-experimental
differences).