Katharina Scheiter

Scheiter

Peter Gerjets
email: p.gerjets@iwm-kmrc.de

Sequence Effects in Solving Knowledge-Rich Problems: The Ambiguous Role of Surface Similarities

Sequence effects are said to occur whenever the problemsolving performance varies as a function of the order in which problems are solved. We present a framework that explain sequence effects as a result of (a) learning during solving a problem and of (b) transferring the learned content on succeeding problems. In two experiments we studied the ambiguous influence of surface similarities among structurally dissimilar knowledge-rich problems on sequence effects. These experiments demonstrate that surface similarities may either foster performance by enabling learning during problem solving or lead to negative transfer depending on whether problem solvers are already aware of the structural features of the problems prior to solving them or not.

Sequence Effects in Problem Solving

. Learning refers to a change in the cognitive system of the problem solver (i.e., newly generated or modified knowledge structures) that occurs due to solving a problem. Transfer refers to the transmission of these newly generated or modified knowledge structures to a subsequent target problem. Whenever problem orders are compared to each other that differ with regard to how they support these two processes, sequence effects may arise.

. In these tasks learning mostly consists in the activation of knowledge structures necessary for the accomplishment of the task and transfer is seen as a more or less automatic process in which higher activated knowledge structures are more readily available when working on a succeeding task.

Sequence effects for knowledge-rich problems These problems differ from knowledge-lean problems in several respects. First, working on knowledge-rich problems requires domain-specific prior knowledge. Second, whereas knowledge-lean problems consist only of a small number of task stimuli that are relevant to their solution (i.e., structural features), knowledge-rich problems are often embedded in a semantically rich context (e.g., algebra word problems). That is, they may contain many surface features that are irrelevant to their solution. As will be stressed in the remainder of this paper, these surface features and their combination with structural features play a central, but ambiguous role for the formation of sequence effects when solving knowledge-rich problems. In particular, surface similarities among succeeding structurally dissimilar problems may either hinder performance by means of negative transfer or improve it by fostering structural awareness and learning during problem solving.

. A situation model is a qualitative representation of a problem in terms of concrete objects, events, and relations mentioned in the problem. The situation model can be transformed into a more abstract and quantitative problem model by further interpreting the situation model on the basis of problem schemas, which are "templates for organizing problem-relevant information" (Nathan et al., 1992, p. 332). These problem schemas are therefore considered to be the relevant prior knowledge helpful to solve knowledge-rich problems. The problem model that results from this knowledge-based interpretation process mainly comprises structural features that determine how a problem has to be solved.

To sum up, learning during solving a knowledge-rich and context-rich problem -as the first factor in the explanation of sequence effects -results in a declarative representation of a problem and its solution that may be

more or less abstract depending on what prior knowledge a problem solver has at his or her disposal. However, there is evidence that even problem schemata themselves are not only abstract, but may include information that is tied to the concrete context of problems [START_REF] Hinsley | From words to equations: Meaning and representation in algebra word problems[END_REF]. In fact, many problem categories (defined by the structural features of the problems) are associated with a specific cover story so that superficial problem features may act as useful cues for retrieving a problem schema from memory. This is, however, helpful only for problems where a cover story activates a problem schema that corresponds to the structural features of the problem. Otherwise, the apparent resemblance with regard to superficial but not structural features may lead to performance impairments.

-Transfer to subsequent problems: Sequence effects are not only based on learning during problem solving but also rely on the fact that the learned content is applied to a new problem. In contrast to knowledge-lean problems, transfer among knowledge-rich problems is said to be a non-automatic process requiring effort and metacognitive awareness [START_REF] Weisberg | Transferring old associations to new situations: A nonautomatic process[END_REF][START_REF] Salomon | Rocky roads to transfer: Rethinking mechanisms of a neglected phenomenon[END_REF]. This kind of transfer is often referred to as transfer by analogy and consists in a number of different processes [START_REF] Gentner | Structure mapping: A theoretical framework for analogy[END_REF][START_REF] Gick | Analogical problem solving[END_REF]. In order to solve a target, first a suitable source problem has to be retrieved from memory. Next, elements of the source problem and its solution are mapped onto the target. Finally, based on these mappings a solution for the target problem can be generated. Whether a target problem is successfully solved by means of analogical transfer depends on first whether a problem solver recognizes similarities between the target and an earlier problem-solving episode stored in memory. Second, this similarity judgment must be based on structural similarity, i.e., it must rely on features that are relevant to the solution of the problems. Unfortunately, a wide range of evidence shows that problem solvers often fail to notice spontaneously that they have encountered similar problems before [START_REF] Gick | Analogical problem solving[END_REF]. Furthermore, similarity judgments are most often based on superficial features of the problems, which sets the stage for negative transfer [START_REF] Holyoak | Surface and structural similarity in analogical transfer[END_REF]. Transfer in solving knowledge-rich problems -as the second factor in the formation of sequence effects -therefore crucially depends upon a problem solver's ability to distinguish among a problem's surface features and its deep structure. Funnily enough, it may be a strong resemblance between problems with regard to superficial features that fosters this structural awareness. That is, comparing two problems that share many surface features but differ with regard to structural features may highlight the structural differences between these problems. This effect has been referred to as the "near-miss" effect [START_REF] Winston | Learning structural descriptions from examples[END_REF].

To conclude, surface features of knowledge-rich problems and the way they are combined with structural features may either hinder performance (negative transfer) or improve it by fostering structural awareness and learning during problem solving (near miss). This ambiguity is expressed in two contrasting hypotheses on sequence effects when solving knowledge-rich problems. These hypotheses can be best explained in the context of the experimental materials that will be introduced next.

Problem Sequences for Algebra Word Problems

The test problems that were used for experimentation consisted in nine algebra word problems. There were three problems that stemmed from each of three problem categories, namely work problems, motion problems, and interest problems [START_REF] Mayer | Frequency norms and structural analysis of algebra story problems into families, categories, and templates[END_REF]. Within each problem category the three problems could be solved by applying the same solution procedure. Different problem categories required different solution procedures. For the purpose of experimentally testing the impact of superficial and structural similarities on sequence effects we made use of a method introduced by [START_REF] Blessing | Content effects in problem categorization and problem solving[END_REF], i.e., we cross-varied the surface and structural features of the word problems (see Table 1).

Motion

Correspond.

Problem category

Interest

Correspond.

Note: Grey-shadowed cells indicate non-corresponding test problems.

For each of the three problem categories there was always one problem that was embedded in the cover story that is typical for problems of that category (i.e., a so-called corresponding problem). Additionally, there were two problems of each problem category that were each couched into cover stories that are typical for another problem category (i.e., non-corresponding problems). For instance, the non-corresponding problem shown in Table 2 belongs to the problem category of work problems, however, its cover story is very similar to that typical of motion problems. In order to investigate the ambiguous role of surface features two different problem sequences were constructed for the nine corresponding and non-corresponding test problems. In the structure-blocked sequence the test problems were presented according to their category membership resulting in three blocks of problems (work-, motion-, interest-category), so that within each block there were three problems that could all be solved by applying the same solution procedure. On the contrary, in the surfaceblocked sequence test problems were presented according to their cover stories resulting in three blocks of superficially similar problems (work-, motion-, interest-story). However, within each block each of the three problems had to be solved by a different solution procedure. In both sequences each block of problems began with a corresponding problem (i.e., a problem couched into its typical cover story), followed by two non-corresponding ones (i.e., problems couched into atypical cover stories).

In the remainder of the paper these problems for which we varied the problem-solving sequence will be referred to as isomorphic problems, because they are structurally identical (isomorphic) to examples participants had to study before working on the test problems. In addition to these isomorphic problems participants had to solve three transfer problems that each required an adaptation of one of the known solution procedures. For the transfer problems order was kept constant for all participants.

We investigated two contrasting hypotheses concerning performance differences between the two problem sequences that are illustrated in the next section.

Transfer versus Near Miss?

The transfer hypothesis states that performance for isomorphic problems in the structure-blocked sequence should be superior to performance in the surface-blocked sequence, whereas the near-miss hypothesis assumes the opposite.

Transfer hypothesis

The first hypothesis is based on an activation-summation model that is commonly used to describe the retrieval of source problems in analogical transfer [START_REF] Holyoak | Surface and structural similarity in analogical transfer[END_REF]. According to this model the retrieval of stored problem knowledge to solve a target problem is guided by structural as well as superficial similarities between target and source. That is, any feature of a target problem can act as a source of activation for mental representations that also contain this feature. "Activation from multiple shared features will summate, and if the activation level of a stored representation exceeds some threshold, that representation will become available for further processing" (Holyoak & Koh, 1987, p. 333f.). The most activated representation is used for solving the current problem. Using this model in order to predict performance for the aforementioned problem sequences leads to the assumption that negative transfer among succeeding problems should occur in the case of a surfaceblocked problem sequence as is illustrated in Figure 1.

The surface-blocked sequence starts with a corresponding problem (indicated by identical indices for problem category PC and cover story CV) that may activate the correct problem schema. The second problem PC2 CV1 is a noncorresponding problem from a different problem category than the first problem, but embedded in a cover story typical for problems of problem category 1. Its structural features may activate the correct problem schema 2; however, its surface features activate problem schema 1, which is already highly activated as it has been applied to solve the first problem. Therefore, problem schema 1 may be spuriously used to solve the second test problem -resulting in negative transfer. A correct problem schema is applied again when the next corresponding problem (PC2 CV2 ) is encountered at the beginning of the second block. On the contrary, in the structure-blocked sequence the structural features of the second problem PC1 CV2 cue the already highly activated problem schema 1 that has been used to solve the first (corresponding) problem and that is also suited to solve the second test problem. Additionally, there may be activation of the wrong problem schema 2 due to shared surface features, however, this should not be sufficient to dominate the selection process. In this problem sequence non-corresponding problems within each block benefit from the previous activation of a problem schema by means of positive transfer, whereas performance for noncorresponding problems is impaired in the case of a surfaceblocked sequence (i.e., negative transfer). According to the transfer hypothesis sequence effects in favor of a structureblocked sequence should in particular be observable for non-corresponding compared to corresponding problems. These sequence effects can be explained by differences among the sequences with regard to enabling positive transfer among succeeding problems.

Near-miss hypothesis According to the second hypothesis a surface-blocked sequence should result in better performance compared to a structure-blocked sequence. This should be the case because a surface-blocked sequence highlights structural differences among succeeding problems as these problems share the same cover story and differ only with regard to a few, but crucial features, i.e. structural features [START_REF] Winston | Learning structural descriptions from examples[END_REF]. On the contrary, in the structure-blocked sequence in which superficially dissimilar problems of one category are presented in succession attention may be directed toward differences with regard to the surface features between these problems and may thereby even impede problem-solving performance. Again, sequence effects should be in particular observable for noncorresponding problems. These sequence effects can be explained by differences among the sequences with regard to enabling learning during problem solving (i.e., acquiring knowledge on structural features). Therefore, the near-miss hypothesis also predicts better performance for transfer problems, if they are solved subsequently to a surface-blocked sequence of isomorphic test problems. Transfer problems require an adaptation of known solution procedures and thus presuppose knowledge on structural features and on their relation to solution steps. The transfer hypothesis, on the other hand, does not allow for any predictions concerning transfer performance.

The two contradicting hypotheses were tested in two experiments described in the following sections.

Experiment 1 Method

Participants Subjects were 40 students (33 female, 7 male) of the Saarland University, Germany, who participated for course credit or payment. Average age was 23.03 years.

Materials and procedure

Participants first received a booklet that contained an example problem together with a step-by-step solution for each of the three problem categories used for experimentation. The worked-out examples were couched into their typical cover stories. Participants were given twelve minutes to study the examples carefully. Subsequently, the example booklet was taken away from them and they received another booklet with the nine test problems that were isomorphic to the previously studied examples. Each test problem was presented on a single page and the pages were arranged according to either the surface-blocked or structure-blocked sequence. Participants were instructed to solve the problems in the given order. There were no time limits during testing. When participants had finished working on the nine isomorphs they had to solve the three transfer problems.

Design and dependent measures

The problem sequence for the nine isomorphic test problems was varied as a between-subjects variable. The problems were either blocked according to the problem category they belonged to (structure-blocked sequence) or according to their cover stories (surface-blocked sequence). The sequence of the three blocks (either containing structurally or superficially similar problems) was counterbalanced within the two experimental conditions. As performance measures subjects' error rates for the isomorphic and transfer problems were registered.

Results and Discussion

In all analyses of variances reported here we used math grades and gender as covariates because they were both strongly associated with problem-solving performance. In a first step we analyzed subjects' performance on the isomorphic test problems by an ANCOVA (Figure 2).

As postulated in the near-miss hypothesis subjects who were presented with the surface-blocked sequence outperformed subjects who worked in the structure-blocked condition (F(1,35) = 4.51; MS E = 305.01; p < .05). Further distinguishing between corresponding and noncorresponding isomorphic problems revealed that this sequence effect in favor of the surface-blocked sequence could only be observed for non-corresponding problems (F(1,35) = 5.27; MS E = 446.94; p < .05), whereas there were no performance differences for corresponding problems as predicted (F < 1). Arranging isomorphic problems according to their cover story compared to arranging them according to their structural features not only improved performance on these problems, but also positively influenced subsequent transfer performance (F(1,34) = 5.17; MS E = 572.02; p < .05). There were no trade-offs between the time needed for problem solving and any of the error measures.

To conclude, performance on the isomorphic as well as on the transfer problems clearly supported the near-miss hypothesis. That is, presenting superficially similar problems successively emphasizes differences between these problems with regard to structural features and thereby fosters problem-solving performance. Additionally, working on problems from different problem categories alternately seems to result in more knowledge on structural features that can be brought into play in order to solve problems that require an adaptation of known solution procedures.

It is important to note that the surface-blocked sequence is superior because it supports a process that seems to be the most crucial prerequisite for transfer, namely recognizing structural features among problems. This explanation has an interesting consequence as it implies that the observed superiority of the surface-blocked sequence should vanish under certain conditions. In particular, there should be no sequence effect due to near miss when problem solvers are already sufficiently aware of the structural properties of the to-be-solved problems.

In order to address this issue we conducted a second experiment where subjects' attention was directed towards structural features of the to-be-solved problems by means of a categorization task before they attempted to solve them either in the structure-blocked or surface-blocked condition.

Experiment 2

Experiment 2 was designed to test the assumption that fostering structural awareness by means of a prior categorization task should reduce the superiority of the surface-blocked sequence. More precisely stated, we expected that participants who were able to categorize the problems according to their structural features would no longer benefit from a surface-blocked sequence in which structural differences are highlighted. On the contrary, it might even well be that due to the fact that good categorizers already know about the problems' structural similarities they might benefit from activation mechanisms in the structure-blocked sequence in which succeeding problems can be solved by the same solution procedure, thereby enabling positive transfer among these problems. However, for subjects who experience difficulties in detecting the structural similarities among problems in the categorization task, the surface-blocked sequence might be another chance to recognize the structural properties of the problems. This superiority of the surface-blocked sequence may however be less accentuated compared to Experiment 1 as it might at least be partially overridden by minor effects of prior problem categorization. In sum, poor categorizers were expected to show a -maybe less accentuatedsuperiority of the surface-blocked sequence, whereas no sequence effect (or even a superiority of the structureblocked sequence) was postulated for good categorizers.

Method

Participants Subjects were 40 students (29 female, 11 male) of the Saarland University, Germany who participated for course credit or payment. Average age was 25.03 years.

Materials and procedure

The same materials were used for experimentation as in Experiment 1. Participants first studied the booklet containing three worked-out examples for a maximum of twelve minutes. In contrast to Experiment 1 they did not start working on the isomorphic test problems immediately after this learning phase. Rather, participants received a sheet of paper that listed all nine isomorphic test problems, whereby the order of presentation depended on experimental condition. They were instructed to categorize problems according to their mathematical features by assigning the letter 'A' to all problems that could be solved in a similar way, the letter 'B' to all problems that could be solved in a different way etc. It was stressed that there could be as many categories as they liked and that the number of problems assigned to each category did not need to be identical for all categories. After having accomplished the categorization task participants were told to solve the problems they had categorized before. The list with the problems to be solved and the category assignments produced by the participant were visible during the whole test phase. Contrary to Experiment 1, the test problems were not presented each on a single page, rather participants received only the list and blank paper to write down their solutions. After having solved the isomorphic problems, participants received the three transfer problems.

Design and dependent measures

Problem sequence for the nine isomorphic test problems was varied as a betweensubjects variable. The problems on the list for the categorization task were either presented in the structureblocked sequence or in the surface-blocked sequence. The sequence of the three blocks (either containing structurally or superficially similar problems) was counterbalanced within the two experimental conditions. Additionally, we distinguished between good and poor categorizers within each of the experimental conditions (see below).

As performance measures participants' error rates for the isomorphic and transfer problems were registered. Additionally, we determined the performance in the categorization task by computing a structure score according to [START_REF] Quilici | Role of examples in how students learn to categorize statistics word problems[END_REF] that indicates participants' ability to categorize problems according to their structural similarities. This structure score was used for a median split within both problem sequence conditions in order to distinguish between good and poor categorizers.

Results and Discussion

In a first step we analyzed subjects' performance on the isomorphic test problems (Figure 3) by means of an ANCOVA (problem sequence x quality of categorization; math grades and gender as covariates). Neither problem sequence nor quality of categorization had an impact on problem-solving performance (problem sequence: F < 1; quality of categorization: F(1,31) = 1.11; MS E = 430.51; p = .30). This lack of significant main effects could be traced back to a cross-interaction between the two factors (F(1,31) = 5.15; MS E = 430.51; p < .05). Participants who performed poorly in detecting structural similarities in the categorization task did not show any sequence effect (t(18) = 0.83; p > .40; two-tailed), whereas good categorizers performed better in the structure-blocked sequence compared to the surface-blocked sequence. However, the latter difference was not significant due to a lack of statistical power (t(18) = -1.51; p < .15; two-tailed).

Further analyses revealed that as expected this pattern of results held only for non-corresponding problems.

For the transfer problems there was no main effect for problem sequence (F < 1). However, good categorizers committed fewer errors in the transfer task than did poor categorizers (F(1,31) = 5.48; M S E = 581.31; p < .05). Finally, there was the same, but less accentuated crossinteraction as for isomorphic problems (F(1,31) = 3.66; MS E = 581.31; p < .10). However, the respective paired comparisons were not significant (both ps > .20).

To conclude, the quality of problem categorization moderated effects of problem sequence as expected. Not only did the sequence effect in favor of the surface-blocked sequence vanish for good categorizers, but participants who had been able to detect the problems' structural similarities in the categorization task even slightly benefited from a structure-blocked sequence. This can be interpreted as evidence that the structural awareness induced by the categorization task supported positive, activation-based transfer among succeeding structurally similar problems in this sequence. On the other hand, for poor categorizers no sequence effect could be observed. It might well be that the initially expected superiority of the surface-blocked sequence is overridden by minor effects of the categorization task that run into the opposite directionadding up to a null effect.

Summary

In two experiments we provided evidence for the ambiguous role of surface similarities in the formation of sequence effects. On the one hand, presenting superficially similar, but structurally dissimilar problems successively can foster learning during problem solving by enabling the recognition of structural relations among problems. On the other hand, structure-blocked sequences may foster the transfer process in the case that structural relations among problems have already been detected. Accordingly, we obtained support for the transfer as well as for the near-miss hypothesis depending on whether problem solvers were aware of structural features prior to problem solving or not.

Figure 1 :

 1 Figure 1: Negative and positive transfer based on activation summation

Figure 2 :

 2 Figure 2: Problem-solving errors for (a) isomorphic problems (total, corresponding and non-corresponding) and (b) transfer problems as a function of problem sequence

Figure 3 :

 3 Figure 3: Problem-solving errors for (a) isomorphic and for (b) transfer problems as a function of problem sequence and quality of categorization

Table 1 :

 1 Algebra word problems used for experimentation

			Cover story
		Work	Motion	Interest
	Work	Correspond.

Table 2

 2

	: Corresponding and non-corresponding test
	problems for the work-problem category
	Corresponding problem: An electrician needs 3 hours in
	order to repair a lamp, while his apprentice needs 6 hours
	to perform the same job. How long would it take them to
	repair the lamp, if they worked together?
	Non-corresponding problem: Jane needs 6 hours to drive to
	Tom by car, whereas Tom needs 4 hours for driving the
	reversed way. How long would they need to drive until
	they meet on the way, if they started at the same time?
	Common solution procedure: 1 = (Rate 1 + Rate 2 ) * h

Acknowledgements

This work was supported by a scholarship of the Deutsche Forschungsgemeinschaft awarded to the first author of the paper as a member of the Graduate College for Cognitive Science at the Saarland University, Germany. We thank Carina Kraemer, Frauke Lancker, Lisa Olbrich, and Tina Schorr for conducting the experiments.


Katharina Scheiter
email: scheiter@iwm-kmrc.de

Peter Gerjets
email: p.gerjets@iwm-kmrc.de

Brigitte Vollmann
email: vollmann_cmr@gmx.de

Richard Catrambone

A Methodological Alternative to Media Comparison Studies: Linking Information Utilization Strategies and Instructional Approach in Hypermedia Learning

Literature reviews on hypermedia learning have yet failed to show consistent positive effects of learner-controlled nonlinear information access. We argue that a possible reason for this lack of evidence in favor of hypermedia learning results from the fact that not sufficient attention is paid to the strategies of information utilization learners deploy. The few studies that do analyze these strategies fail to link them to an instructional approach, which hampers a deeper interpretation of strategy patterns. Our study showed that different groups of learners can be distinguished according to their strategies used in an example-based hypermedia environment and that these groups differ with regard to learning outcomes, but not individual learner characteristics.

Is Learning with Hypermedia Ineffective?

Hypermedia environments are nonlinear information networks, which can be explored in multiple ways with the learner having control over the selection, sequencing, and pacing of information. Learner control is seen as beneficial for knowledge acquisition for different reasons: First, learners can adapt the presentation of information to their preferences and needs. Second, learner control requires a deeper information processing, because learners have to compare the options offered to come to an informed decision. Third, learner control may train meta-cognitive self-regulation abilities. Fourth, it may improve motivation to learn and learners' attitude towards the topic. Despite these hypothesized advantages, literature reviews have yielded ambiguous results concerning the effectiveness of hypermedia learning over other forms of instruction [START_REF] Dillon | Hypermedia as an educational technology: A review of the quantitative research literature on learner comprehension, control, and style[END_REF].

In the remainder of the paper, two arguments will be brought forward and supplemented by some initial empirical support: First, it is proposed that analyzing the way learners make use of information offered in hypermedia environments might be a promising methodological alternative to global media comparisons. Most of the existing studies tend to ignore that once learner control is provided, learners' information utilization behavior will show an increased variance. Thus, whether hypermedia is effective will depend on whether learners use the available information in a way that knowledge acquisition processes are facilitated.

Second, the analysis of information utilization strategies should be linked to the instructional approach implemented in a hypermedia environment. Based on cognitive task analyses and empirical evidence, an instructional approach (e.g., learning from worked-out examples) can be characterized by a set of cognitive processes (e.g., example elaborations and comparisons) that enable knowledge acquisition (e.g., construction of a problem schema), where different cognitive processes require different information as input. Thus, an effective information utilization strategy can be defined a priori by the fact that information is selected, sequenced, and paced in a way that cognitive processes relevant to the instructional approach are facilitated.

Until now, there are only a few studies that have considered strategies of information utilization when studying hypermedia learning. [START_REF] Barab | Understanding kiosk navigation: Using log files to capture hypermedia searches[END_REF] used students' navigational profiles based on different strategy indicators (e.g., number of pages retrieved, depth of search) to predict whether a user had been given a specific information search goal for browsing the system. [START_REF] Balcytiene | Exploring individual processes of knowledge construction with hypertext[END_REF] identified different groups of learners (i.e., self-regulated learners, cue-dependent learners) by means of logfile and video analyses for her hypertext system on gothic architecture. Cue-dependent learners did not browse the environment in a systematic way and oriented themselves towards its local aspects, while self-regulated learners reflected on the accessed information and were oriented towards the global structure of the system. In accordance with this interpretation of the navigational profiles and video data, the posttest revealed superior retention for self-regulated learners. [START_REF] Lawless | Understanding hypertext navigation through cluster analysis[END_REF] used a cluster-analytical approach to identify different groups of users (i.e., knowledge seekers, feature explorers, and apathetic users), which were then compared with regard to different external criteria. Knowledge seekers, who had used the information most intensively, had a higher prior knowledge and showed better learning outcomes than the other groups. The authors conclude that "the navigational strategies employed by the 'knowledge seekers' are the most like the sophisticated reading strategies used by competent traditional text processors" (p. 395f.). Moreover, apathetic users may lack the prior knowledge necessary to deploy effective information utilization strategies.

The aforementioned studies take a first step into the right direction by focusing on users' information utilization strategies. However, their interpretation of learners' navigational profiles is done in a post-hoc way without linking them to the instructional approach implemented in the hypermedia environment (if there is one approach to begin with). In the next section, we illustrate how an analysis of information utilization strategies can be based on an instructional approach and its underlying cognitive processes.

Defining Information Utilization Strategies for an Example-based Hypermedia Environment

For our study we chose the instructional approach of example-based learning to illustrate the claim that research on hypermedia learning can be improved by analyzing learners' strategies of information utilization in terms of the instructional approach implemented in the hypermedia system: The advantage of choosing example-based learning is that there are already many findings on cognitive strategies supporting this instructional approach -many of them being based on cognitive load theory (CLT, [START_REF] Sweller | Instructional design in technical areas[END_REF].

The CLT is an instructional design theory that specifies helpful conditions for cognitive skill acquisition. These skills are assumed to be represented as problem schemas. Many of the instructional settings analyzed on the basis of cognitive load theory involve the use of worked-out examples for conveying problem schemas, as examples have been shown to be very successful for this purpose at least for novice learners (cf. [START_REF] Atkinson | Learning from examples: Instructional principles from the worked examples research[END_REF]. CLT assumes a direct causal relationship between a specific instructional design, cognitive activities, the resulting pattern of cognitive load, and the learning outcomes is assumed. Recently, Gerjets and colleagues [START_REF] Gerjets | When are powerful learning environments effective? The role of learner activities and of students' conceptions of educational technology[END_REF][START_REF] Gerjets | Goal configurations and processing strategies as moderators between instructional design and cognitive load: Evidence from hypertext-based instruction[END_REF] have extended the CLT so that it can be used to analyze learner-controlled settings, where learners can select among different strategies of handling the instructional materials. In the augmented CLT, it is thus proposed that whether an instructional design results in either helpful or harming cognitive load depends on learners' strategies of information utilization. Moreover, learner characteristics are included as factors that may influence strategy selection. The augmented CLT may thus serve as a framework for analyzing information utilization strategies in example-based hypermedia environments.

From a cognitive load perspective, effective information utilization strategies with respect to examples consist in selecting examples that facilitate cognitive processes relevant to the acquisition of problem schemas. Examples that support schema acquisition have the following characteristics: First, effective examples reduce the intrinsic load inherent to the domain (cf. [START_REF] Gerjets | Designing instructional examples to reduce intrinsic cognitive load: Molar versus modular presentation of solution procedures[END_REF]. Second, examples that foster schema acquisition keep extraneous cognitive load, which results from cognitive processes not relevant to learning, at a minimum. Third, they facilitate higher-level cognitive processes that go beyond the mere activation of information in working memory, and that result in germane cognitive load. These processes consist in example elaborations and comparisons. Elaborations occur when learners draw inferences concerning the structure of example solutions, the rationale behind solution procedures, and the goals that are accomplished by individual solution steps (i.e.. self-explanations, [START_REF] Renkl | Learning from worked-out examples: A study on individual differences[END_REF]. Beyond self-explanations, learners should engage in example comparisons in order to notice structural features that differ among problem categories and that are shared by problems within a category [START_REF] Quilici | Role of examples in how students learn to categorize statistics word problems[END_REF].

Based on these characteristics, four groups of example utilization strategies were identified: Learners should retrieve examples that (1) result in low intrinsic cognitive load, (2) support processes of comparison (3) stimulate selfexplanations, (3) and compensate for lacking selfexplanations. In the following, these strategies will be illustrated in the domain of probability theory, where extensive research has been done on the differential effectiveness of different example formats in system-controlled settings. In the hypermedia environment used in the study all these example formats were included and learners could select among formats that had been proven either effective or noneffective in the prior system-controlled studies.

Low Levels of Intrinsic Cognitive Load

A low intrinsic cognitive load has been found for modular rather than molar examples (Gerjets et al., 2004;[START_REF] Gerjets | Can learning from molar and modular worked-out examples be enhanced by providing instructional explanation and prompting self-explanation[END_REF]. Molar examples have a recipe-like structure and refer to complex entities like problem categories, clusters of structural task features, and category-specific solution procedures. In modular examples, solution procedures are broken down into smaller meaningful groups of solution steps that can be understood in isolation. They require learners to keep only a limited number of elements active simultaneously in working memory. Evidence for the superiority of modular examples was found in terms of learning time, self-reported cognitive load, and later problem-solving performance for isomorphic and novel problems (Gerjets et al., 2004[START_REF] Gerjets | Can learning from molar and modular worked-out examples be enhanced by providing instructional explanation and prompting self-explanation[END_REF]. Accordingly, an effective example utilization strategy consists in preferring modular over molar examples.

Supporting Example Comparisons

Comparing examples with different surface features within problem categories and comparing examples with similar surface features across categories are both helpful cognitive processes to identify the relevant structural problem features of problem categories [START_REF] Scheiter | When less is sometimes more: Optimal learning conditions are required for schema acquisition from multiple examples[END_REF]. Comparing examples with different surface features within categories may help to identify varying features, which must be irrelevant for category membership, while commonalities may indicate structural features. On the other hand, comparing examples with the same surface features across problem categories may highlight structural differences among the examples. In general, differences among instances become more salient, the more other features are shared by them. Thus, for comparisons examples should be selected that differ with regard to only a few features, which can then be identified and interpreted more easily.

Stimulating Self-explanations

Self-explanations are an important aspect of meaningfully processing examples although they seldom occur spontaneously [START_REF] Renkl | Learning from worked-out examples: A study on individual differences[END_REF]. Self-explanations can be fostered by presenting incomplete examples whose gaps need to be filled in [START_REF] Paas | Training strategies for attaining transfer of problem-solving skill in statistics: A cognitiveload approach[END_REF], by presenting prompts that ask learners to generate self-explanations [START_REF] Berthold | Fostering the understanding of multi-representational examples by selfexplanation prompts[END_REF], or by a combination of both methods [START_REF] Atkinson | Transitioning from studying examples to solving problems: Combining fading with prompting fosters learning[END_REF]. However, in our own studies we could not find any beneficial effects of prompting learners to give an explanation for why a specific step to solve a probability problem had been selected [START_REF] Gerjets | Can learning from molar and modular worked-out examples be enhanced by providing instructional explanation and prompting self-explanation[END_REF]. In fact, while self-explanation prompts did not affect learning from molar examples, they even hindered learning from modular examples. We explained these results by assuming that learners in the modular-examples condition were forced to generate self-explanations for material that they had already sufficiently understood and that was thus redundant to them. Accordingly, one might argue that at least for those learners who have already understood the principles illustrated, an effective example utilization strategy might consist in not retrieving examples that contain self-explanation prompts. On the other hand, it might well be that if learners can decide by themselves whether to select these examples, only learners who need processing support will retrieve them.

Compensating for Lacking Self-explanations

Students often overestimate their understanding of examples and thus refrain from further elaborating them. Moreover, even if they have noticed gaps in their knowledge, they may not be able to generate self-explanations to overcome those gaps. These problems may be solved by providing additional instructional explanations, particularly for learners with low prior knowledge. Doing so can show learners that they suffer from an illusion of understanding and may help them to overcome comprehension difficulties. Thus, an effective example utilization strategy for novices would be to retrieve examples with instructional explanations. However, as explanations sometimes do not affect learning [START_REF] Gerjets | Can learning from molar and modular worked-out examples be enhanced by providing instructional explanation and prompting self-explanation[END_REF] or are even harmful because they hinder learners in generating explanations by themselves [START_REF] Aleven | Limitations of student control: Do students know when they need help?[END_REF], learners should retrieve them only if they cannot produce explanations by themselves.

To conclude, effective example utilization strategies con-sist in retrieving modular examples, in comparing examples that differ only in a few, but relevant features, and in finding the right balance between using examples with or without self-explanation prompts and explanations depending on their level of understanding. This also implies that strategy patterns should be analyzed rather than investigating the effects of single strategy variables in isolation. This is why a cluster-analytical approach was preferred over conducting correlational analyses.

Study Method

Participants. Seventy-six students of the University of Tuebingen, Germany, were paid to participate in the study.

Average age was 25.0 years (32 male, 44 female).

Materials and Procedure. We used a hypermedia environment that taught learners how to calculate the probability of complex events. It consisted of an example-based learning phase, and a test phase with problem-solving tasks, and a declarative knowledge test. In the learning phase, each of four problem categories was explained by two worked examples, which differed with regard to their cover story: Always one example dealt with selecting marbles from an urn, while the other was related to daily-life situations. Thus, the same surface features were used across categories for the urn examples, but varied for the daily-life examples.

To access an example, learners first had to select one of the problem statements from the left navigation bar (Figure 1). The problem statement was then displayed on the format-selection page together with eight links that allowed retrieving different formats for the presentation of the solution procedure. These formats varied with regard to the solution approach by offering either molar or modular structured solution procedures and with regard to the degree of elaboration and completeness of examples. tion steps, but no further explanations were given, while in low-elaborated examples only the mathematical information was given.Moreover, incomplete examples with selfexplanation prompts / feedback could be selected. Here the solution procedure was medium elaborated and learners were prompted to provide the missing explanations by themselves. After they had typed in the explanation, a feedback page appeared, which contained the learner's response and the system-provided expert explanation. This procedure had to repeated until explanations had been given for all solution steps; only then another example could be selected. In total, learners could choose among 64 options (i.e., 8 problem statements x 8 solution formats). All participants received the same options to retrieve examples. Learners could decide on their own when to enter the test phase.

Analyses and Measures. The analyses consisted in three steps: First, the students' example utilization strategies were subjected to a cluster analysis to identify learner groups with distinct strategies. Second, the relationship between group membership and learning outcomes was investigated. Third, it was tested whether the groups differed with regard to individual learner characteristics.

The variables used to describe example utilization strategies were the overall example study time, the overall frequency of retrieving examples, the frequency of retrieving either modular or molar examples as well as the frequency of retrieving either highly-elaborated, medium-elaborated, low-elaborated or incomplete examples. Moreover, we assessed the time spent on the format-selection page as a potential indicator for meta-cognitive awareness. Beyond these measures for the selection and pacing of information, sequencing activities were registered by Markov-chain analyses. The resulting variables comprised the number of transitions between the examples' cover stories, the average number of dimensions changed within a transition from one example to another (e.g., '1' for only changing the degree of elaboration vs. '4' for changing the cover story, solution approach, degree of elaboration, and the category in parallel), and the number of transitions between the molar and modular solution approach. These variables were supposed to provide information on learners' comparison strategies.

The dependent measures consisted in overall performance for solving three isomorphic and six novel problems and in the declarative knowledge test. For each of the nine test problems as well as for the eleven items of the declarative posttest one point was assigned for a correct answer; no partial credit was given. Moreover, we assessed individual learner characteristics, which were expected to be associated with strategy selection. These included a questionnaire on cognitive and meta-cognitive strategies in mathematics [START_REF] Wolters | Advancing achievement goal theory: Using goal structures and goal orientations to predict students' motivation, cognition, and achievement[END_REF], the Epistemological Beliefs Instrument [START_REF] Jacobson | Epistemological beliefs instrument: Scales and Items[END_REF], the Attitudes Towards Mathematics Inventory [START_REF] Tapia | An instrument to measure mathematics attitudes[END_REF]) and items to assess the preference for amount of instruction [START_REF] Hannafin | Preferences and learner control over amount of instruction[END_REF]. The original questionnaires were slightly modified and shortened in order not to overwhelm students. Prior knowledge was assessed by domain-unspecific and specific indicators (i.e., final high school grade, math grade, and performance in a declarative pretest, which was the same as the declarative posttest).

Results

To identify groups of learners that differ in their example utilization strategies, a cluster analysis (based on the Ward algorithm) was performed on the strategy variables. The so called 'elbow-criterion' was used to stop the clustering process after four clusters had been identified. A threeclusters solution would have increased the within-group variance for the new cluster substantially and thus would have resulted in a loss of information.

For the four-clusters solution, the groups of learners differed significantly in their example utilization strategies (with the exception of 'number of transitions between the cover stories' and 'number of dimensions changed within one transition', cf. Table 1). Cluster 1 -called the unreflective-intermediate example users -spent less time on selecting examples than students of Clusters 3 and 4. They studied the selected examples more intensively than students in Clusters 2 and 3, but less than students in Cluster 4. The longer overall example time was the only variable that distinguished between Cluster 1 and 2. Cluster 2 -the unreflective-lazy example users -did invest less time on reflecting on the appropriateness of the different formats for displaying the solution procedure and on processing examples than any of the other clusters. Moreover, students in Cluster 2 refrained from switching between modular and molar examples compared to students in Clusters 3 and 4 and did not control for possible illusions of understanding compared to Cluster 4 students. Cluster 3 -the reflective-sparse example users -seemed to make well thought-over decisions regarding the solution procedure's format. Learners in this group processed an intermediate number of examples comparable to that of students in Cluster 1, but used less time to study them. Cluster 4 -the excessive example users -spent less time on the format-selection page than students of Cluster 3, but retrieved the most examples, which were moreover extensively processed. In particular, they differed from the other groups in their frequent use of molar and of incomplete examples.

In a second step, problem-solving performance and performance in the declarative knowledge test were analyzed by an ANOVA using group membership as a betweensubjects factor. Unreflective-intermediate example users and unreflective-lazy example users both performed poorly in the problem-solving task, while reflective-sparse example users and excessive example users performed rather well. These differences across the four groups were significant (F(3,72) = 5.13; p < .01), while post-hoc comparisons highlighted differences between Clusters 1 and 3, and Clusters 2 and 3, respectively. Conducting the same ANOVA for the declarative knowledge test revealed slightly different results (F(3,72) = 3.72; p < .05). Unreflective-intermediate example users achieved 76.19% correct in the test, which was less then for the reflective-sparse and excessive example users. There were no other differences among the groups. Contrary to our expectations, there were no differences between the four groups with regard to individual learner characteristics with the exception of the final high school grade (F(3,72) = 3.02; p < .05). Reflective-sparse example users had better high school grades than unreflectiveintermediate example users.

Summary and Discussion

We were able to demonstrate that the analysis of information utilization strategies that have been defined with respect to a specific instructional approach implemented in a hypermedia learning environment helps to identify groups of learners with distinct strategy patterns. These strategy patterns can account for differences in subsequent problemsolving performance as well as in declarative knowledge acquisition. Interestingly, more than one effective strategy pattern could be identified: First, an excessive example utilization proved to be successful, even though these students preferred examples with a high intrinsic cognitive load (i.e., molar examples). One possible explanation for the latter might be that these students at the same time most frequently monitored their understanding by selecting incomplete examples. Thus, they might not have chosen the most efficient strategy -which would have been to focus on modular examples -but compensated for that by a very thorough example processing. Second, learners also performed well when using examples only sparingly. In the latter case, the display format for the solution procedure seemed to have been rather carefully selected as indicated by the time spent on the format-selection page. Thus, studying only a few examples can be an effective strategy as long as one invests more thinking on selecting the most suitable examples (i.e., meta-cognitive effort). Moreover, the time spent on the format-selection page might also be interpreted as a learner's attempt to reason about a possible solution to the problem, before looking at a specific solution procedure (cf. anticipative reasoning, [START_REF] Renkl | Learning from worked-out examples: A study on individual differences[END_REF]. Interestingly, learners with this example utilization strategy had better average high school grades than at least one of the other groups. This might indicate better general abilities that help to regulate one's learning activities in an efficient way. Contrary to results by [START_REF] Renkl | Learning from worked-out examples: A study on individual differences[END_REF] or Lawless and Kulikowich (1997), learners with this efficient, but probably knowledge-rich strategy formed a large group in our sample.

While the pacing and selection variables were suited to distinguish among different behavioral patterns, the sequencing variables only accounted for a small part of the variance. These variables were problematic, because they were defined at a very coarse level making it hard to interpret them unambiguously. For instance, looking only at examples from different problem categories embedded in the same cover story might be as effective as repeatedly comparing examples within a problem category embedded into different cover stories -both strategies resulting in completely different values for the respective variable. Thus, in the future a more fine-grained definition and analysis of sequencing strategies needs to be deployed.

Unfortunately, although we assessed a variety of learner characteristics, we could not identify characteristics that differentiated between different example utilization groups. We used constructs that have been identified as being relevant to learner-controlled instruction in the literature from a theoretical perspective and we selected the most reliable measures available to assess these constructs. Thus, there might be something wrong with questionnaires that assess cognitive and meta-cognitive knowledge and attitudes in general. Accordingly, [START_REF] Winne | Methodological issues and advances in researching tactics, strategies, and self-regulated learning[END_REF] have suggested using unintrusive data like logfiles not only to analyze strategic behavior, but also to interpret this data as indicators for meta-cognitive abilities. Moreover, most of the questionnaires that have been used to predict behavior in learner-controlled settings assess global cognitive and metacognitive abilities rather than domain-specific ones.

Beyond these methodological difficulties, analyzing the way learners make use of the information provided seems to be a more fruitful approach to investigating hypermedia than global media comparisons. The latter fail to consider the specific characteristics of the media and thus produce ambiguous results. Analyses of learners' information utilization may furthermore help to improve instruction by informing the design of strategy prompts and trainings to support students in learning with hypermedia.

Figure 1 :

 1 Figure 1: The format-selection page Highly-elaborated examples provided detailed explanations for why a solution step had been chosen. Mediumelaborated examples mentioned facts concerning the solu-

Table 1 :

 1 Example utilization and performance in the four clusters

				Means			Results of the post-hoc Tukey tests
			CL 1	CL 2	CL 3	CL 4	CL 1 -	CL 1 -	CL 1 -	CL 2 -	CL 2 -	CL 3 -
			(n = 21)	(n = 12)	(n = 34)	(n = 9)	CL 2	CL 3	CL 4	CL 3	CL 4	CL 4
	Time (sec)
	Format selection **	1.95	1.83	8.30	5.33	ns	**	*	**	*	*
	Overall	**	11.12	3.20	8.02	25.89	**	**	**	**	**	**
	Frequencies
	Overall	**	15.00	10.33	19.24	31.11	ns	ns	**	**	**	**
	Molar	**	7.62	4.50	8.91	16.56	ns	ns	**	ns	**	**
	Modular	*	6.62	5.67	9.88	13.44	ns	ns	ns	ns	(*)	ns
	Highly-elab.	*	8.38	3.00	7.00	10.89	ns	ns	ns	ns	*	ns
	Medium-elab.	**	1.90	1.00	5.15	7.44	ns	(*)	*	(*)	*	ns
	Low-elab.	*	1.57	5.75	5.68	6.22	ns	*	ns	ns	ns	ns
	Incomplete	*	3.14	0.58	1.41	6.56	ns	ns	ns	ns	**	**
	Transitions
	Cover story	ns	1.48	1.42	1.41	1.44	ns	ns	ns	ns	ns	ns
	Dimensions	ns	1.53	1.58	1.50	1.45	ns	ns	ns	ns	ns	ns
	Approach	*	2.33	1.00	6.18	9.56	ns	*	**	*	**	ns
	Performance (%)
	Problem-solving		29.87	25.67	50.80	48.48	ns	*	ns	*	ns	ns
	Declarative knowledge	76.19	84.09	86.10	98.90	ns	*	*	ns	ns	ns
	High school grade	2.29	2.28	1.87	2.20	ns	*	ns	ns	ns	ns
	Note: ns = not significant; ** = p ≤ .01; * = p ≤ .05; (*) = p ≤ .10.