Inductive Reasoning - SAGE Journals

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Review of Educational Research March 2008, Vol. 78, No. 1, pp. 85–123 DOI: 10.3102/0034654307313402 © 2008 AERA. http://rer.aera.net

Inductive Reasoning: A Training Approach Karl Josef Klauer Technical University of Aachen, Germany Gary D. Phye Iowa State University, Ames Researchers have examined inductive reasoning to identify different cognitive processes when participants deal with inductive problems. This article presents a prescriptive theory of inductive reasoning that identifies cognitive processing using a procedural strategy for making comparisons. 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 review and meta-analysis summarizes the results of 74 training experiments with nearly 3,600 children. Both hypotheses are confirmed. Further, two moderating effects were observed: Training effects on intelligence test performance increased over time, and positive problem-solving transfer to academic learning is greater than transfer to intelligence test performance. The results cannot be explained by placebo or test-coaching effects. It is concluded that the proposed strategy is theoretically and educationally promising and that children of a broad age range and intellectual capacity benefit with such training.

KEYWORDS: cognitive training, inductive reasoning, problem-solving transfer. 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, Ray, & Kamhi, 1993), 85

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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 on process models that aim to solve certain kinds of problems 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 reported here by stressing principles such as similarities and diversities. In contrast to most of the research mentioned, the prescriptive theory described in the next section 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 participants 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 beyond the scope of experience by asserting something about a nonobserved or even nonobservable universe of objects. Drawing inductive inferences is much more demanding but also much more critical than the 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 because a number of experiments have been performed by 86

Inductive reasoning consists of detecting regularities and irregularities by finding out A

B

a1 similarity a2 difference

b1 attributes of

a3 similarity &

b2 relations

difference C c1 verbal c2 pictorial with

c3 geometrical

material.

c4 numerical c5 other

FIGURE 1.

Definition of inductive reasoning.

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) 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” (p. 200). 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 or (b) the detection of a rule, or is it characterized by (c) 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). According to Figure 1, inductive reasoning reveals not only 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 87

TABLE 1

Types of inductive reasoning problems Process

Facet Item identification formats

Cognitive operation required

Generalization

a1b1

Similarity of attributes

Discrimination

a2b1

Cross classification

a3b1

Recognizing relationships

a1b2

Differentiating a2b2 relationships System construction a3b2

Class formation Class expansion Finding common attributes Identifying disturbing items 4-fold scheme 6-fold scheme 9-fold scheme Series completion Ordered series Analogy Disturbed series Matrices

Discrimination of attributes (concept differentiation) Similarity and difference in attributes Similarity of relationships Differences in relationships Similarity and difference in relationships

Note. See Figure 1 for the facet identification definitions.

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 because attributes are identified as predicates with one argument, whereas relations are identified as predicates with two or more arguments. Because 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 are varieties of classification tasks, whereas 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 88

SYSTEM CONSTRUCTION

CROSS CLASSIFICATION

GENERALIZATION

DISCRIMINATION RECOGNIZING

DIFFERENTIATING

RELATIONSHIPS Similarity

Difference

RELATIONSHIPS

Similarity

Attributes

Difference

Relationships

STRATEGY OF INDUCTIVE REASONING

FIGURE 2.

Genealogy of tasks in inductive reasoning.

to solve these items. The relationships among the six basic varieties of inductive reasoning tasks are depicted in the genealogy of Figure 2. 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 attributes and the relationships 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), to find out commonalities and/or diversities. Presumably, a computer program could be developed to solve any problem of inductive reasoning. 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. 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 89

Start

Compare the objects (pairs of objects) globally Try again Build a hypothesis

Test the hypothesis by directed comparisons

Rule discovered? yes

End FIGURE 3.

no

Already tried several times ?

no

yes Apply the analytical strategy

Heuristic or hypothesis-guided strategy of inductive reasoning.

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 90

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modify it appropriately for the six varieties of inductive tasks, and to experience sufficient opportunities through practice 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 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 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 (Klauer, 1989, 1991, 1993): • Program I for children age 5 through 8, • Program II for children age 11 through 13, and • Program III for youths age 14 through 16.

Program I is basically nonverbal so that it can be used in all language environments. For this reason, an American version (Klauer & Phye, 1994) and a Dutch version (Klauer, Resing, & Slenders, 1996) of the manual have been published. 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 youths with weak performances in school and who are at risk for vocational integration. Training Format Each of the programs, I, II, and III, consists of 120 items, that is, 20 items for each of the six basic classes of inductive reasoning tasks. Further, Programs II and III offer 40 verbal, 40 figural, and 40 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 because 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 10 lessons. Beginning with Lesson 2, the metacognitive aspects are at the center of attention. In Lessons 2 and 3, problem classes are defined by attributes, whereas in Lesson 4 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 5 to 7 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. 91

TABLE 2

Instructional objectives of the 10 lessons Lesson Training objective

Comment

1

Solving the problems naively

2

Distinguishing attributes and relationships

3

Recognizing the three attribute classes Recognizing the three relationship classes

The children should get familiar with the material and the training situation Introduction of the terms attribute and relationship; sorting all items of Lesson 1 appropriately Distinguishing the three classes; sorting all of the attribute items thus far Distinguishing the three classes; sorting all of the relationship items thus far; recapitulation of the attribute problems Learning how to solve and check generalization and recognizing relationship problems; recapitulating sorting of items Learning how to solve and check discrimination and differentiation of relationship problems; recapitulating sorting of items Learning how to solve and check crossclassification and system construction problems; recapitulating sorting of items Rehearsing all of the processes with attribution problems Rehearsing all of the processes with relation problems Practicing all types of identifying, solving, and checking processes with all types of problems

4

5

Solving and checking procedures with similarity problems

6

Solving and checking procedures with difference problems

7

Solving and checking procedures with similarity and difference problems Repeating and practicing problems of the attribution branch Repeating and practicing problems of the relation branch Mixed repetition of all kinds of problems and procedures

8 9 10

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 differences.” 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? 92

Answer Similarity or difference or both with attributes or relationships. Compare (i.e., look for similarity or difference or both). I do it according to an assumption or systematically. By the opposite comparison.

Inductive Reasoning Training

(In the response to Question 3, 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 (see Appendix A) have been performed where at least one 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 meta-analysis to gain an overview of the most important results. Based on the prescriptive theory of inductive reasoning and a review of the research literature cited, it is possible to derive certain hypotheses that can be tested meta-analytically using the available database. 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—for example, the Cattell Culture Fair Tests (Cattell & Cattell, 1963) or the 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 that 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 93

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another year in a kindergarten-like environment in the school but without exposure to the three Rs. For instance, these mildly learning disabled or otherwise exceptional children may be less responsive to 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 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 children 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 the master’s degree. Although 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 94

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(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 and 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) crosssectional research, in which 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 (the coaching hypothesis). Hager et al. (2000) have suggested the possibility that the positive training effects can be the result of teaching to the test or to coaching procedures. Coaching to a test is an old and widespread practice, particularly where test performance determines admission to certain careers 95

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(Anastasi, 1981). However, participants in Klauer’s training programs are instructed to solve six item classes that are found in many intelligence tests. During training, participants are taught not only to solve specific items but a general strategy of compare and contrast that 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, whereas 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 hypotheses. 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 that 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. 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. Only three of the experiments in the data pool (see Appendix A) were not published, the studies by Brünken (No. 56), Bussas (No. 7), and Favre (No. 55). These experiments were theses and were not published mainly because they did not add new insights or results to the body of research that was available at that time. Method For a study to be included in the meta-analyses testing Hypotheses 1 through 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) 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 to 3 focus on transfer to intelligence test performance. The central hypothesis (Hypothesis 1) predicts transfer of inductive reasoning training to performance on intelligence tests that 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). 96

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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 noninductive 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 design of two training groups without a notraining 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 no-training control group to the two training groups. If both treatments yield more or less the same effects, comparisons with the no-treatment control group enable 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 noninductive training programs were used—training of spatial cognition with the program Tetris (Masendorf, 1994; Souvignier, 1997), training of metacognitive strategies using problem solving with noninductive problems (Bornemann, 1989; Klauer, 1992; Kolmsee, 1989), arithmetical training (Angerhoefer, Kullik, & Masendorf, 1992), training in reading strategies (Klauer, 1996), training in other academic learning strategies and 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 5 weeks. As a rule, participants were tested a week before training and a week after training (pretest-posttest design), resulting in a total training study duration of 7 weeks. A few of the training studies needed less time, whereas others needed more time, depending on the given circumstances. All of the training experiments used such a pretest-posttest design. In addition, 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 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 the Raven Coloured Progressive Matrices, the Standard Progressive Matrices, or the Advanced Progressive Matrices. Others included subtests of inductive reasoning items, such as the Cattell Culture Fair Tests, the German version of the Cognitive Ability Test, or the German version of the Columbia Mental Maturity Test for learning disabled children. In some studies, 97

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only those subtests that require inductive reasoning were administered, whereas other studies included the complete test. Thus, the data on which the meta-analysis is based refer to intelligence tests that in part, or in their entirety, measure inductive reasoning. Furthermore, 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. Hypothesis 4 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 no-training control group. Most training groups participated in inductive training for two sessions a week over a 5-week period, whereas 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 groups together. This second phase was experienced as a regular classroom activity by students in both the experimental group and the control group. As a rule, after choosing an appropriate subject matter, an informal criterion-referenced 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, and 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 through 7. Numbers 8 through 13 deal with cognitive variables, which do not include learning of a subject matter. The verbal descriptors for variables coded 1 through 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 pretestposttest increase for the training and control groups. Research Design Shortcomings The overwhelming majority of the training experiments included only small samples of children or youths because it is more efficient to conduct inductive reasoning training in small groups rather than in an entire class. This is a trade-off. Although 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 sample size 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. 98

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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 g=

MT – MC sp

Because we are often dealing with small sample sizes and because pretest means can vary considerably in this case, even when the participants 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 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: 99

TABLE 3

Frequency of contrasts used in the meta-analysis Type of contrasta

Frequency of experimentsb

Frequency of effect sizes

1 only 2 only 1+3 1+1 1+2+3 1+1+3 Sum

49 13 6 (39, 33, 23, 8, 5, 3) 3 (64, 34, 22) 1 (73) 2 (52, 9) 74

49 13 12 6 3 6 89

Commentc Nonproblematic Nonproblematic Nonproblematic Problematic Problematic Problematic

a. 1 = inductive training versus no training control group; 2 = inductive training versus another but not inductive training; 3 = another training versus no training control group. b. The numbers in parentheses refer to experiment reference numbers in Appendix B. c. 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.

d = (1 3 ) 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 do studies with smaller samples. This procedure generally leads to lower mean effect sizes. Hedges and Olkin also developed correcting procedures to account for a lack of test reliability. Because applying these corrective procedures would lead to higher means, corrections for attenuation were not employed. 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. 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 labeled Contrast and denoted by 1, 2, or 3. Here are the explanations: Contrast 1: An inductively trained group is contrasted with a no-training control group. Contrast 2: An inductively trained group is contrasted with an alternatively trained group. Contrast 3: An alternatively trained (non–inductive reasoning training) group is contrasted with a no-training control group. 100

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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 reasoning 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. 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 criterionreferenced 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 Contrast 1 types 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 6 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 has the following ramifications: (a) the d values and the corresponding means receive lower estimations, (b) d values are corrected for (larger) sampling errors, and (c) the means are based on 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 70 because no intelligence test was administered, 73 experiments are available where the effect of inductive training on intelligence test performance can be ascertained. Based on 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 = 3,595). 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 shows a rather symmetrical distribution. As expected, there is considerable variability among the effect sizes. Some are quite small, whereas 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 = .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 101

FIGURE 4. Distribution of the g effect sizes of training transferring to intelligence test performance.

outliers, the effect sizes are normally distributed. Regardless, the following analyses are based on 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 previously mentioned unweighted mean g (Mg = 0.59). Nevertheless, one can conclude that on average inductive training improves 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 < .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 homogeneous 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 are the type of students involved, the training conditions, and the two varieties of 102

TABLE 4

Weighted means d+ of the effects of the inductive training on intelligence test performance: Summary of meta-analyses Variable

d+

95% confidence interval

n

N

p(Q)

All experiments Possible moderator variables Programs Program I Program II Program III

0.52

0.46–0.59

79

3,595

.80

0.57 0.43 0.50

0.49–0.66 0.32–0.55 0.31–0.67

42 24 13 79

2,004 1,144 447 3,595

.65 .63 .99

Subjects Kindergarten School kindergarten Primary school Secondary school Special education

0.47 0.43 0.61 0.42 0.54

0.24–0.70 0.11–0.75 0.50–0.72 0.30–0.54 0.39–0.69

9 5 19 24 22 79

306 153 1,274 1,148 714 3,595

.09 .95 .63 .78 .98

Training conditions One-to-one Pairs Small groups Classes

0.34 0.59 0.57 0.51

0.15–0.53 0.41–0.78 0.46–0.67 0.40–0.62

11 15 36 17 79

432 466 1,422 1,275 3,595

.70 .64 .84 .50

Authorship Staff Klauer Other authors

0.57 0.49

0.47–0.68 0.40–0.57

35 44 79

1,386 2,209 3,595

.79 .80

Note. n = number of contrasts; N = number of subjects; p(Q) = probability of the coefficient Q of homogeneity.

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. Hypothesis 2 (placebo effect) assumes that the effect of inductive training is due simply to the special social conditions of any training. 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 with 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. 103

TABLE 5

Weighted means d+ of the effects of the inductive training on learning: Summary of meta-analyses Variable

d+

95% confidence interval

n

N

p(Q)

All experiments Possible moderator variables Programs Program I Program II Program III

0.69

0.59–0.79

38

1,723

.35

0.64 0.64 0.84

0.49–0.80 0.49–0.79 0.62–1.06

11 16 11 38

663 698 362 1,723

.89 .87 .04

Subjects Primary school Secondary school Special education

0.63 0.59 0.94

0.47–0.80 0.43–0.75 0.74–1.14

8 15 13 36

594 650 434 1,678

.71 .90 .20

Training conditions Small groups Classes

0.73 0.62

0.60–0.86 0.46–0.79

22 11 33

969 612 1,581

.38 .76

Authorship Staff Klauer Other authors

0.67 0.71

0.54–0.81 0.57–0.86

20 18 38

966 757 1,723

.79 .80

Note. n = number of contrasts; N = number of subjects; p(Q) = probability of the coefficient Q of homogeneity.

Hypothesis 2 is tested by comparing the alternatively trained group with the notraining control group (i.e., Contrast 3). 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? To address this question, 22 experiments involving 1,094 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 Appendix B, column labeled Months). 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 = .44 (df = 19), p = .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. 104

FIGURE 5. Distribution of the g effect sizes of training transferring to academic learning performance.

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 = 1,723) 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. Figure 5 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. 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 < .05), which is in contrast to expectations. Obviously, inductive reasoning training improves academic learning of schooltype subject matter more than it improves measured intellectual functioning. Moreover, with p(Q) = .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, on closer inspection, one moderator variable effect requires additional consideration. Within Program III, 11 experiments did not produce homogeneous effect sizes, p(Q) = .04. The largest outlier effect sizes on learning were found in studies with older students in special education who were trained using Program III. Prior to analyzing the social conditions of the training, 5 experiments are excluded from the analyses because of too few participants (three studies with 105

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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 9 experiments in which this issue was specifically addressed lead 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 that have been reported in educational settings (Adair, Sharpe, & Huynh, 1990) and in 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 that 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 on 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. 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 106

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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 that 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 10 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 hypothesis of teaching to the test, or coaching (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 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 participants actually proceed this way when they solve inductive problems. Instead, our contention is only that participants 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 levels, 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), in which 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 107

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that a strategy that 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 of compare and contrast is very concrete, and teachers should be able to adapt it to their regular lessons instead of relying on published training programs. This way, they would have the opportunity to adapt the strategy according to their students’ levels of cognitive development and 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. APPENDIX A Meta-analyses data pool

73.-74. Sonntag, W. (2004). Experimentelle Untersuchungen zum Einfluss des Klauerschen Denktrainings auf mathematisches Denken und Lernen von lernbehinderten Sonderschülern [Experimental studies on the impact of Klauer’s training to reason on mathematical thinking and learning with slightly retarded students]. Zeitschrift für Pädagogische Psychologie, 18, 101–111. 70.-72. Sonntag, W. (2002). Fördert ein Training des induktiven Denkens das Lösen mathematischer Textaufgaben? [Does a training of inductive reasoning improve solving of mathematical word problems?]. Heilpädagogische Forschung, 28(1), 24–37. 68.-69. Roth-van der Werf, T. J. M., Resing, W. C. M., & Slenders, A. P. A. C. (2002). Task similarity and transfer of an inductive reasoning training. Contemporary Educational Psychology, 27, 296–325. 67. Klauer, K. J., Willmes, K., & Phye, G. D. (2002). Inducing inductive reasoning: Does it transfer to fluid intelligence? Contemporary Educational Psychology, 27, 1–25. 66. Möller, J., & Appelt, R. (2001). Auffrischungssitzungen zur Steigerung der Effektivität des Denktrainings für Kinder I [Booster sessions to foster the effectiveness of the training to reason for children I]. Zeitschrift für Pädagogische Psychologie, 15, 199–206. 65. Sydow, H., & Schmude, C. (2001). Training des analogen Denkens und des Zahlbegriffs im Vorschulalter [Training of analogical reasoning and of the number concept with preschoolers]. In K. J. Klauer (Ed.), Handbuch kognitives Training [Handbook of cognitive training] (pp. 129–164). Göttingen, Germany: Hogrefe. 64. Koning, E. de (2000). Inductive reasoning in primary education: Measurement, teaching, transfer (chap. 6). Dissertation Universität of 108

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Utrecht. See also Koning, E. de, Hamers, J. H. M., Sijtsma, K., & Vermeer, A. (2002). Teaching inductive reasoning in primary education. Developmental Review, 22, 211–241. 63. Fries, S. (2001). Ein Training zur gleichzeitigen Förderung des Leistungsmotivs und des induktiven Denkens [A training to foster simultaneously achievement motivation and inductive reasoning]. Münster, Germany: Waxmann. 62. Hager, K., Hübner, S., & Hasselhorn, M. (2000). Zur Bedeutung der sozialen Interaktion bei der Evaluation kognitiver Trainingsprogramme [On the impact of social interaction on the evaluation of cognitive training programs]. Zeitschrift für Pädagogische Psychologie, 14, 106–115. 61. Braun, J., Weyhreter, H., Köhnlein, O., Storck, M., & Bode, H. (2000). Kognitives Training: Ein Programm zur Förderung von Vorschulkindern mit intellektuellen Defiziten [Cognitive training: A program to foster preschoolers with intellectual deficits]. Psychologie in Erziehung und Unterricht, 47, 10–17. 60. Klauer, K. J. (1999). Induktives Denken oder elementares Wahrnehmen? Ein Entscheidungsexperiment [Inductive reasoning or perception?]. Empirische Pädagogik, 13, 97–122. 58-59. Strathmann, A. (1999). Denktraining bei Lernbehinderten: Transferiert es auf Intelligenz und Lernen? [Training to reason with learning disabled subjects: Does it transfer on intelligence and learning?]. Heilpädagogische Forschung, 25, 129–139. 56.-57. Brünken, R. (1993). Auswirkungen eines Denktrainings für Kinder auf intentionales und inzidentelles Lernen bei Lesetexten [Effects of a training to reason on intentional and incidental learning with instructional texts]. Unpublished master’s thesis, University of Aachen, Germany. 55. Favre, M. (1994). Der Einfluss eines Trainings des induktiven Denkens auf die Leistung im intentionalen und inzidentellen Lernen bei schulischen Lehrtexten [On the influence of an inductive training on intentional and incidental learning with instructional texts]. Unpublished master’s thesis, University of Aachen, Germany. 54. Strathmann A. (1999). Über die Effekte eines Strategietrainings bei verhaltensgestörten Schülern und Regelschülern [On the effects of a strategy training with behavior disturbed and regular students]. Psychologie in Erziehung und Unterricht, 46, 177–186. 53. Möller, J. (1999). Denktraining für Jugendliche: Homogenität der Trainingsgruppen und Booster-Sessions [Training to reason for youths: Homogeneity of training groups and booster sessions]. Heilpädagogische Forschung, 25, 2–7. 52. Fries, S., Lund, B., & Rheinberg, F. (1999). Lässt sich durch gleichzeitige Motivförderung das Training des induktiven Denkens optimieren? [Can training of inductive reasoning be optimized through fostering simultaneously achievement motivation?]. Zeitschrift für Pädagogische Psychologie, 13, 37–49. 51. Langfeldt, H.-P., & Schlieper, J. (1999). Aspekte der konvergenten und diskriminanten Validität des “Denktrainings für Kinder I” von K. J. Klauer [Aspects of convergent and discriminant validity of the “Training 109

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to Reason for Children I” by K. J. Klauer]. Psychologie in Erziehung und Unterricht, 46, 1–6. 49.-50. Klauer, K. J. (1999). Über den Einfluß des induktiven Denkens auf den Erwerb unanschaulich-generischen Wissens bei Grund- und Sonderschülern [On the impact of inductive reasoning on the acquisition of abstract-generic knowledge with primary and special school students]. Psychologie in Erziehung und Unterricht, 46, 7–28. 48. Tomic, W., & Kingma, J. (1998). Accelerating intelligence development through inductive reasoning training. In W. Tomic & J. Kingma (Eds.), Conceptual issues in research on intelligence (pp. 291–305). Stamford, CT: JAI. 47. Souvignier, E. (1998). Effekte kognitiver Trainingsprogramme der Vorstellungsfähigkeit und des induktiven Denkens auf die Problemlösefähigkeit [Effects on problem solving ability of cognitive training programs to foster spatial performance and inductive reasoning]. Zeitschrift für Experimentelle Psychologie, 45, 20–28. 46. Klauer, K. J. (1998). Begünstigt induktives Denken den Erwerb der Gedächtnisstrategie des Kategorisierens? [Does inductive reasoning improve the acquisition of the memory strategy of categorizing?]. Zeitschrift für Pädagogische Psychologie, 12, 73–84. 44.-45. Hamers, J. H. M., De Koning, E., & Sijtsma, K. (1998). Inductive reasoning in third grade: Intervention promises and constraints. Contemporary Educational Psychology, 23, 132–148. 43. Möller, J., & Köller, O. (1997). Effekte von Leistungsgruppierung und Ausgangsfähigkeit auf die Wirksamkeit des Denktrainings II [Effects of grouping according to achievement level and of pre-experimental ability on the effectiveness of the Training to Reason II]. Zeitschrift für Entwicklungspsychologie und Pädagogische Psychologie, 29, 242–254. 42. Resing, W. C. M., & Slenders, A. P. A. (1997, September). Training inductive thinking skills in learning disabled children. Paper presented at the 8th European Conference on Developmental Psychology at Rennes, France. In W. Resing & W. Tomic (Co-convenors), Teaching higher order thinking skills. 41. Rollett, W., & Wladika, E. (1997, August), The effect of Klauer’s inductive reasoning training. A consequence of trainer-trainee interaction? Paper presented at the 7th European Conference on Learning and Instruction, Athens. In I. Diakidou (Chair), Problem solving. 40. Klauer, K. J. (1997). Lässt sich die Strategie des induktiven Denkens auf schulisches Lernen transferierbar lehren? [Can the strategy of inductive reasoning be taught to transfer on learning in school?]. Zeitschrift für Entwicklungspsychologie und Pädagogische Psychologie, 29, 225–241. 39. Souvignier, E. (1997). Die Förderung des räumlichen Denkens bei lernbeeinträchtigten Schülern [Fostering spatial reasoning with learning impaired students]. In F. Masendorf (Ed.), Experimentelle Sonderpädagogik [Empirical special education theory] (pp. 379–420). Weinheim, Germany: Deutscher Studien Verlag. 37.-38. Tomic, W., & Klauer, K. J. (1996). On the effect of training inductive reasoning: How far does it transfer and how long do the effects persist? European Journal of Psychology of Education, 11, 283–299. 110

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35.-36. Klauer, K. J. (1996). Begünstigt induktives Denken das Lösen komplexer Probleme? [Does inductive reasoning improve solving of complex problems?]. Zeitschrift für Experimentelle Psychologie, 43, 85–113. 34. Klauer, K. J. (1996). Teaching inductive reasoning: Some theory and three experimental studies. Learning and Instruction, 6, 37–57. 33. Klauer, K. J. (1996). Denktraining oder Lesetraining? Über die Auswirkungen eines Trainings zum induktiven Denken sowie eines Lesetrainings auf Leseverständnis und induktives Denken [Training to reason or training to read? The effects of an inductive reasoning training and a reading training on reading comprehension and inductive reasoning]. Zeitschrift für Entwicklungspsychologie und Pädagogische Psychologie, 28, 67–89. 31.-32. Beck, M., Lübking, M., & Meier, U. (1995). Die Bielefelder Studien zum Denktraining von Klauer [Bielefeld studies on Klauer’s training to reasoning]. In W. Hager (Ed.), Programme zur Förderung des Denkens bei Kindern (pp. 294–308). Göttingen, Germany: Hogrefe. 30. Klauer, K. J. (1995). Induktives Denken erleichtert die Konstruktion analoger Satzstrukturen [Inductive reasoning improves the construction of analogical sentence structures]. Sprache & Kognition, 14, 221–227. 28.-29. Klauer, K. J. (1995). Weitere Erprobung des “Denktrainings für Jugendliche” in der Oberstufe der Schule für Lernbehinderte [Another test of the “Training to Reason for Youths” in special education]. Heilpädagogische Forschung, 21, 157–170. 27. Hasselhorn, M., Hager, W., & Boeley-Braun, K. (1995). Läßt sich die fluide Intelligenz erwachsener Behinderter durch das Aachener Denktraining nachhaltig verbessern? [Can fluid intelligence of handicapped adults be sustainably improved by the Aachen Training to Reason?]. Heilpädagogische Forschung, 21, 171–179. 26. Hager, W., & Hasselhorn, M. (1995). Zuwendung als Faktor der Wirksamkeit kognitiver Trainings für Kinder [Attention as a factor of the effectiveness of cognitive training programs for children]. Zeitschrift für Pädagogische Psychologie, 9, 163–179. 25. Tomic, W. (1995). Training in inductive reasoning and problem solving. Contemporary Educational Psychology, 20, 483–490. 24. Hasselhorn, M., & Hager, W. (1995). Neuere Programme zur Denkförderung bei Kindern: Wie effektiv sind sie im Vergleich zu herkömmlichen Wahrnehmungsübungen? [Newer programs to foster children’s reasoning: How effective are they in comparison to traditional perceptual training?]. Psychologie in Erziehung und Unterricht, 42, 221–223. 23. Masendorf, F. (1994). Förderungstypen des induktiven Denkens und des räumlichen Vorstellens bei lernbeeinträchtigten Kindern [Types of improvers of inductive reasoning and of spatial ability with learning impaired children]. Psychologie in Erziehung und Unterricht, 41, 14–21. 22. Klauer, K. J. (1994). Transferiert der Erwerb von Strategien des induktiven Denkens auf das Erlernen eines schulischen Lehrstoffs? [Does the acquisition of a strategy of inductive reasoning transfer on learning of school type subject matter?]. Zeitschrift für Pädagogische Psychologie, 8, 15–25. 111

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Schmude, C., & Sydow, H. (1994, September). Wirkungen kognitiven Trainings im Vorschulalter [Effects of cognitive training programs with preschoolers]. Contribution at the 39th Congress Deutsche Gesellschaft für Psychologie, Hamburg (Section: Evaluation of programs to foster cognitive development). 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 Prädiktions-KFA [Fostering thinking and math performance with learning impaired children: Multivariate evaluation of change by prediction CFA]. Psychologie in Erziehung und Unterricht, 39, 190–195.

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8.

7. 6.

5.

4.

3.

2.

1.

Klauer, K. J. (1992). Problemlösestrategien im experimentellen Vergleich: Effekte einer allgemeinen und einer bereichsspezifischen Strategie [Comparing problem solving strategies experimentally: Effects of a general and a domain specific strategy]. In H. Mandl & H.F. Friedrich (Eds.), Lern- und Denkstrategien [Learning and thinking strategies] (pp. 58–78). Göttingen: Verlag für Psychologie. 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, Germany. 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 I [Training to reason for children I] (p. 64). Göttingen, Germany: Verlag für Psychologie. 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 I [Training to reason for children I ] (pp. 64-65). Göttingen, Germany: Verlag für Psychologie. Johnen, M. (1989). Erprobung des Denktrainings im Kindergarten [Testing the Training to Reason in kindergarten]. In K. J. Klauer, Denktraining für Kinder I [Training to reason for children I ] (p. 75). Göttingen, Germany: Verlag für Psychologie. Bornemann, K. (1989). Erprobung des Denktrainings in Zweiergruppen [Testing the Training to Reason with pairs of children]. In K. J. Klauer, Denktraining für Kinder I [Training to reason for children I ] (pp. 73–74). Göttingen, Germany: Verlag für Psychologie. Bornemann, K. (1988). Erprobung des Denktrainings im Kindergarten [Testing the Training to Reason in kindergarten]. In K. J. Klauer, Denktraining für Kinder I [Training to reason for children I ] (p. 73). Göttingen, Germany: Verlag für Psychologie. Bornemann, K. (1988). Erprobung des Denktrainings im Kindergarten [Testing the Training to Reason in kindergarten]. In K. J. Klauer, Denktraining für Kinder I [Training to reason for children I ] (pp. 72-73). Göttingen: Verlag für Psychologie.

113

114

Experiment

74 73 73 73 72 71 70 69 68 67 66 65 64 64 63 62 61 60 59 58 57 56 55 54

Row

89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 69 68 71 70 67 66

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0

Author

N

30 39 38 20 36 29 40 79 31 166 54 62 91 97 156 32 18 48 24 24 46 25 31 30

APPENDIX B Summary table of meta-analyses

5 5 5 5 5 5 5 3 5 3 3 3 3 3 4 3 1 4 5 5 4 2 4 4

Subject 1 1 2 3 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 2 1 1

Contrast 3 3 3 3 3 2 2 1a 1a 1 1 1 1a 1a 2b 1 1 2 3 1 2 1 3 3

Program 3 3 3 3 3 3 3 2 2 3 3 3 4 4 4 1 1 4 3 1 4 2 3 3

Social condition 0.64* 0.25 0.49 –0.22 0.49* 1.06* 0.24 0.22 0.90* 0.83* 0.81* 0.69* 0.59* 0.27* 0.05d 0.33 0.35* 0.83* 0.47 0.39* 0.71* 0.83* 0.59*

4 4 4 2 2 1 1 2c 2c 2 2 5 5 2 4 5 5

g11

2 2 2 2 5 5

DV 1

1.05* 0.34 1.22* 0.73* 0.23

0.73* 0.88* 0.59*

5 15 15 4

12 3 3

0.80*

g12

6

6

Months

2 4 4 11 13 8 3 2 2 7 2 4 4

2 2 2 2 2 2 2 11 11 2

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

0.84* 1.68* 1.50* –0.28 0.82* 1.33* 0.62* 0.16 1.31* 0.81*

g2

(continued)

DV 2

115

Experiment

53 52 52 52 51 50 49 48 47 46 45 44 43 42 41 40 39 39 38 37 36 35 34 34

65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42

0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1 1 1 1

Author

(continued)

Row

APPENDIX B

40 51 29 39 24 54 40 47 40 41 331 39 28 32 40 48 29 29 28 43 60 84 22 22

N

5 4 4 4 1 3 4 3 5 3 3 3 4 5 4 4 5 5 3 3 4 4 3 3

Subject 1 1 1 3 1 2 2 1 1 2 1 1 1 1 1 1 1 3 1 1 1 2 1 1

Contrast 3 2b 2 2 1 1 1 1 2 1 1 1 2 1a 2 2 3 3 1 1 2 2 1 1a

Program 3 4 4 4 2 3 2 4 3 2 4 4 3 2 3 3 4 4 3 3 4 4 3 3

Social condition 2 2 2 2 3 4 7 4 5 2 1 5 2 8 2 6 5 5 4 2 3 6 2 2

DV 1 0.46* 0.50* 0.09 –0.14 1.24* 0.30 0.49* 0.67* 0.83* 0.46* 0.56* 1.00* 0.42* 0.76* 0.60* 0.48* 0.45* 0.26 1.13* 0.39* 1.13* 0.31* 0.76* 0.59*

g11

0.83*

1.51* 0.54*

9 4

0.54* 0.50*

g12

4

5

7 4

Months

(continued)

0.80* 0.18 0.74* 0.42*

0.53* 0.96*

4 7 2 2 1 1

0.70

0.46* 1.24* 0.47* 1.07* 0.75*

g2

11

5 9 2 1 9

DV 2

116

Experiment

33 33 32 31 30 29 28 27 26 25 24 23 23 22 22 21 20 19 18 17 16 15 14 13

41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18

1 1 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1

Author

(continued)

Row

APPENDIX B

22 22 68 60 32 36 45 28 34 34 32 20 20 35 34 20 30 140 32 44 51 32 36 61

N

4 4 4 5 4 5 5 5 2 5 2 5 5 4 4 1 2 1 2 4 4 5 5 4

Subject 3 1 1 1 1 2 2 2 2 1 2 1 3 1 1 1 2 1 2 1 1 1 1 1

Contrast 2 2 2 1 2 3 3 2 1 1 1 1 1 2a 2a 1 1 1 1 2 2 3 3 2

Program 3 3 3 1 3 3 3 1 2 3 1 3 3 4 4 1 1 1 1 3 3 2 4 3

Social condition 5 5 2 2 2 5 5 2 2 2 2 2 2 6 6 2 2 3 2 2 2 5 2 6

DV 1 –0.17 0.80* –0.05 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*

g11 –0.18 1.09*

0.39 0.71*

0.89* 0.51*

0.34

6 4

3 5

10

g12

6 6

Months

1.00* 0.89* 0.35 1.11* 0.65*

3 5 2 2 6

(continued)

0.21 1.00* 1.01* 1.13* 0.56* 0.97* 0.04

9 12 12 3 3 2 9

0.36

0.60* 1.37*

5 7

8

0.73* 0.48* 0.16

g2

5 5 10

DV 2

117

Experiment

12 11 10 9 9 9 8 8 7 6 5 5 4 3

17 16 15 14 13 12 11 10 9 8 7 6 5 4

1 1 1 0 0 0 1 1 1 1 1 1 1 1

Author

(continued)

Row

APPENDIX B

16 30 24 15 15 20 20 20 56 20 20 19 19 22

N

1 4 4 5 5 5 3 3 4 3 3 3 1 1

Subject 1 1 1 1 1e 3 1 3 1 1 1 3 1 1

Contrast 1 2 2 1 1 1 1 1 2 1 1 1 1 1

Program 1 3 3 3 3 3 2 2 4 3 2 2 2 2

Social condition 4 6 6 2 2 2 2 2 5 3 3 3 4 4

DV 1 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*

g11

4 7

Months

0.99* 0.34*

g12

2

0.56*

g2

(continued)

DV 2

118

3 2 1

Experiment

1 1 1

Author

(continued)

22 20 27

N

1 1 1

Subject 3 1 1

Contrast 1 1 1

Program 2 2 2

Social condition 4 4 3

DV 1 0.26 0.54* 1.12*

g11 7

Months 0.39*

g12

DV 2

g2

*p ≤ .05. a. Modified version of the program. b. Combination of the inductive training and a motivational training. c. Cattell Culture Fair Tests, Subtests 3 to 5, that is, the inductive subtests only. d. dpost instead of dcorr. e. Contrasted to a different control group. Legend for the columns of Appendix B: Row is the number of the contrast. Experiment is the number of the experiment in Appendix A. Author 1 = experiment was performed by students or staff members of Klauer. Author 0 = experiment was performed by other authors. N is the number of participants. Subject is the kind of subjects: 1 = kindergarten, 2 = school kindergarten, 3 = primary school, 4 = secondary school (including 9 experiments in grammar schools), 5 = special education. Contrast is the type of contrast: 1 = inductive training versus no training control group, 2 = inductive training versus another but not inductive training, 3 = another training versus no training control group. Program is the number of the program: 1 = Program I, 2 = Program II, 3 = Program III. Social condition is the social condition of the training: 1 = one-to-one training, 2 = training of pairs of children, 3 = training in small groups of 3 to 5 children, 4 = training of intact classes. DV 1 is the first dependent variable: 1 = informal test of inductive reasoning, 2 = Cattell Culture Fair Tests (German version), 3 = Cognitive Abilities Test (German version), 4 = Coloured Progressive Matrices, 5 = Standard Progressive Matrices, 6 = Advanced Progressive Matrices, 7 = Columbia Mental Maturity Test for mildly retarded children (German version), 8 = another intelligence test. g11 is the effect size g of Dependent Variable 1 immediately after training (corrected for preexperimental differences). Months is the time interval between g11 and g12. g12 is the effect size g of Dependent Variable 1 the stated number of months after g11 (corrected for preexperimental differences). DV 2 is the 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 noninductive intelligence test, 12 = test of spatial reasoning, 13 = Visual Discrimination Test POD. g2 is the effect size g of Dependent Variable 2 (corrected for preexperimental differences).

3 2 1

Row

APPENDIX B

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Authors KARL JOSEF KLAUER is professor emeritus of education at the Institute of Education, Technical University of Aachen, Germany; e-mail [email protected] He is especially interested in the research of teaching and learning. GARY D. PHYE is professor of curriculum/instruction and psychology in the College of Human Sciences of Iowa State University, N162b Lagomarcino, Ames, IA 50011-3190; e-mail [email protected] He is especially interested in problem-solving transfer and academic learning.

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