Setting the Right Example

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Setting the Right Example

Rasila Hoek

August 12, 2015

Thesis Msc. Mathematics Supervision: dr. A.J.P. Heck

Korteweg-De Vries Instituut voor Wiskunde

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Abstract

In education research it is claimed that Studying worked examples is an effective, well-tested method for acquiring mathematical skills, which has been found to be most ef-fective in the beginning stage of learning a new mathematical technique, before starting with practice problems. This timing principle can be violated in short courses where many new skills have to be acquired in a short time. First-year Dutch university students in psychobiology, taking a short course in calculus, have been observed while they solve problems and study worked examples at the same time. What problems occur when students solve problems relying on worked examples, and how can these problems be solved?

Thinking aloud experiments with nine psychobiology students demonstrated that the students had a strong tendency to rely heavily on only one worked example while solving a problem in calculus. A classification of the errors that these students made pointed at ineffective copy-and-adapt strategies, related to insufficient or incorrect self-explanation of the steps in the worked example and to a lack of comparison between the worked example and the problem.

In previous research, the addition of self-explanation prompts and meaningful building blocks to a worked example have been claimed to be effective techniques for supporting students in studying worked examples. A combination of the two techniques was tested in an experiment with fourteen prospective students in economy and finance, taking a short course in basic calculus. Most students had difficulties providing correct and useful self-explanations and did not take into account important differences between the problem and the worked example that they used.

It seems that students in the beginning stage of skill acquisition need more support in providing self-explanations and in comparing the worked examples and the prob-lem. Recommendations for future research topics include the use of other types of self-explanation prompts, stimulating students to use more than one example, and inserting theory in worked examples.

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Acknowledgments

I would like to thank Andr´e Heck for being an enthusiastic supervisor and for taking his time to introduce me thoroughly to the field of education research. Secondly, I am grateful to my sister Punya, who shares my passion for education and teaching, and helped me structuring my thoughts countless times. Also, my special thanks go to Freek, for encouraging me to go on, step by step. And finally, this research would not have been possible without both the psychobiology students and the summer school students, thanks to whom collecting the data was not only useful, but also lots of fun.

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1 Introduction

I start with a vignette of a student in a tutorial class, learning the method of substitution in integration. She is struggling with finding the following indefinite integral:

Z

x(2x − 5)6dx

After a few minutes, the student asks the teacher to check if the steps so far are correct, and shows a piece of paper with the following calculations:

u = 2x − 5 du = dx

∫ x(2x − 5)6dx = ∫ 1₂u +5₂ u6du

The teacher starts to check the calculations, concludes there is a mistake in the second step, and asks the student to check the derivation of 2x − 5 again, assuming a simple mistake in computing the derivative has been made. The student is confused and asks what exactly is her mistake.

Actually nothing is wrong with this student’s ability to differentiate. The mistake this student made is caused by something else, something that is key to this research. In this introduction I describe the context of my research, explain why mistakes as the above are made, and describe how the classroom observation of such mistakes led to the research topic and research aims.

1.1 Clinging to worked examples

The context of this research is calculus education for beginning university students in the Netherlands, in particular for students who do not study mathematics as their main subject. There are two groups of students involved in the research, to which we will refer as the psychobiology group and the summer school group:

• First-year students of the bachelor in Psychobiology at the University of Amster-dam, taking a course in calculus during the second semester.

• Prospective students in economics and finance, taking a course in basic algebra and calculus during the month of July.

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The characteristics of both courses will be discussed in more detail in Chapter 4. How-ever, for now I would like to stress some important similarities between the two groups that are key to understanding the motivation of the research purpose. These important similarities concern three aspects:

• The goal of the course is to equip the student with mathematical tools that are necessary to use in his or her field of study.

• Correspondingly, the type of mathematics that is taught is mainly procedural: students learn techniques in calculus, like differentiating a trigonometric function or solving an ordinary differential equation.

• The course is given in a short period (ten and four weeks for the psychobiology and the summer school group, respectively).

Designing and teaching a short course focusing on mathematics as a tool at university level is challenging for several reasons. Firstly, a balance has to be found between giving background, perspective, and explanation on the one hand, and describing techniques to solve problems on the other hand. Even though the focus is on the use of mathematics as a tool, some background, mathematical reasoning, or even proofs may be necessary. Secondly, the course should enable students to learn a large amount of new mathematical techniques in a relatively short time.

During eight weeks of exploratory observance of the psychobiology students during tuto-rials, a clear picture arose of how students, in particular the weaker students, deal with the time pressure and the large amount of new mathematical concepts and techniques: they cling to worked examples. Even with access to a lecture, instructional texts, pic-tures, formulas, theorems, derivations, explanations and proofs, when asked to compute an integral in a weekly test, students resort to the worked example that, in their opinion, resembles their test question most. Though this behavior is understandable and maybe also wise in many cases, this approach can lead to problems and mistakes as well. The student described in the first part of this introduction, was a psychobiology student, and used the worked example shown in Figure 1.1.

The example was well chosen by the student, since the integrals are indeed similar, and the idea to choose the part between brackets as u was transferred in the correct way from the worked example to the problem that has to be solved. However, the student interpreted the step du = dx as something that is apparently ‘some standard formula’ to be used in substitution, rather than something that has to be computed, and decided to just imitate the example in this. It is clear that the student has a lack of understanding as for the meaning of the differential notation, and needs to either review the theory on this subject, get some explanation of the teacher, or study some additional worked examples of substitution, to conclude that u has to be differentiated with respect to x in order to find du.

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Figure 1.1: A worked example of integration by the method of substitution

Though it is probably easy to correct the student, the mistake points at an important issue concerning worked examples as main learning source: When is a worked example, or a series of worked examples, sufficient for the student to derive the general technique and solve a slightly different problem? To what extent is it possible to design worked examples such that transfer from the worked example to the problem is facilitated?

Given the goal of both the psychobiology course and the summer school course (learning calculus in order to use it as a tool in other fields), a key role of worked examples seems appropriate: the desired outcome is that a student has learned to apply certain mathematical techniques in the correct way and in the right situation. Several previous studies have reported that worked examples can have a leading role in the acquisition of mathematical skills (see, for example, Zhu & Simon, 1987; Carrol, 1994 ; Renkl, 2014). However, it seems that students need to be guided as for how, when and to what extent they can use worked examples. Possibly, this guidance can partly lie in the way worked examples are chosen, structured, formulated, or put into context. This idea, arisen from the observed behavior of the psychobiology students, has been the driving force for my research aims.

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1.2 Research topic and research aims

The main subject of this research is the use of worked examples in the setting of short calculus courses. A driving characteristic of these courses is the time pressure, which seems to lead to the typical learning behavior of solving problems relying on worked ex-amples. The focus of this research is on this particular way of using worked exex-amples.

The purpose of this research is to contribute to the field of worked example research, and, more specifically,

• to get a clear and precise idea of how students use worked examples during problem solving in an introductory calculus course;

• to identify the problems that can arise when students work like this, and;

• to develop some recommendations on how to prevent or solve these problems.

Though much research has been done on the question in which phase worked examples should be used to maximize learning gains, in practice (and especially in situations of time pressure or self-directed learning) these principles of timing are sometimes hard to abide to: students seem to have their own way and moment of using worked examples. The relevance of this research lies in observing the effect of worked examples in prac-tice, and in possibly adjusting or adding principles of design and structuring of worked examples in order to suit the setting of short calculus courses.

1.3 Structure of the thesis

In the next chapter, I review research literature on the subject of worked examples, discuss the important concepts in the field and summarize the relevant findings so far. In Chapter 3, I present my research questions and describe the research design. In Chapter 4, the details concerning the educational setting are provided: the two student groups and the courses are described. In Chapter 5, I elaborate on the analysis framework and I present the results following from this analysis in Chapter 6. Finally a discussion of the results, including recommendations for future research is given in Chapter 7.

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2 Theoretical Framework

Learning by examples, that is, studying worked examples of a mathematical problem in order to learn a certain solving strategy, is an instruction method that has been subject of research for several decades. The idea is as follows: after being introduced to the basics of a certain mathematical formula, principle, or algorithm, students are provided with a number of worked examples to study, before starting with practice problems. In my research, I looked at a slightly different situation, in which problem solving and studying worked examples go hand in hand. However, the research on the classical worked example method is highly useful as a background: firstly, it provides insight into the benefits of studying worked examples, also in comparison to problem solving; secondly, it offers explanations why studying worked examples can increase learning gains; and thirdly, it gives rise to various instructional principles that are important when using worked examples. In this chapter, I review the most relevant literature written on the subject and discuss the key concepts in the field.

2.1 Introduction to the worked example research

Learning by worked examples has been studied in various contexts, ranging from mathe-matics and physics to debating and argumentation. I restrict my literature review to the studies dealing with well-structured domains such as mathematics and physics, which are most relevant to this research. Four types of research papers on the subject can be distinguished, based on the type of research question discussed in the paper.

Firstly, much research has been done concerning the effectiveness of learning by worked examples. Important here is the way in which effectiveness is defined and measured. A frequently used measure is the performance of the student when solving problems that are largely comparable to the worked example. In order to check if students do not merely make a simple syntactic mapping (that is, replace numbers) between the worked example and the test problem, often more complex test problems are used as well, as to judge the transfer capability of the student, that is, the extent to which a student can transfer the procedure of the worked examples to other, more complex problems.

Secondly, research has been done focusing more specifically on transfer issues. What are the reasons that worked examples are in some cases insufficient to enable the student to solve a problem that is more complex than or a little different from the example? Is it possible to increase this transfer capability? An important concept brought up in this second type of research is self-explanation. Self-explanation is the activity of explaining the steps in the worked example to oneself, either aloud or mentally. Questions that

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are dealt with in the research on explanation, are of the following kind: Does self-explanation enhance learning, and in what way? Is it useful to stimulate students to self-explain aloud? What is the relation between self-explanation and transfer capability? Other studies with the aim to find a solution for transfer problems focus on instructional principles such as meaningful building blocks and example sets.

Thirdly, there are articles that deal with timing issues: in what phase of skill acquisi-tion is studying worked examples most effective, and why? An important concept from cognitive load theory that is discussed here is the knowledge-gain reversal effect, which describes the decreasing effectiveness of studying worked examples during the process of skill acquisition. In order to provide an instructional principle that accounts for this effect by supporting the transition from studying worked examples to solving problems, the technique of fading worked-out steps is introduced.

Lastly, the structuring of worked examples is an important topic. How should the worked examples be structured in order to yield the best learning results? How to deal with different sources of information, such as pictures, equations and text? Concepts from cognitive theory and multimedia learning, such as cognitive load and split-attention are important here. An instructional principle presented in this type of research papers is the easy-mapping principle.

2.2 Effectiveness of learning by worked examples

Renkl (2014) claimed that learning by worked examples is considered a well-tested, successful method for acquiring problem-solving skills, if implemented in the right way, that is, if certain instructional principles are taken into account. In this section I briefly discuss some of the first studies in which the effectiveness of learning by worked examples was empirically researched. I also summarize the factors that influence the effectiveness of learning by worked examples.

Studies on effectiveness

Zhu and Simon (1987) were apparently the first who claimed effectiveness of learning by worked examples, doing empirical research on learning algebra at high school level. They found that students who are only provided with worked examples and practice problems perform at least as good as students provided with an instructional text and practice problems. Most pupils who were provided with the worked examples could apply and slightly adjust the solving procedure in situations different from the provided exam-ples, in other words, some transfer was possible. Moreover, studying worked examples significantly increased the speed of skill acquisition.

These conclusions were supported by the research of Carrol (1994). In addition, he observed that students provided with worked examples in combination with practice problems showed a tendency to start with the practice problem and use the worked ex-ample during the work on the practice problem. This happened even if the pupils were instructed to first study the worked examples. This is similar to the ‘clinging to the

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example’ observed in the psychobiology group, as described in the introduction.

Other early studies pointed at difficulties of learning by worked examples, too. Reed (1985) conducted classroom research at college level, using examples of algebra word problems. He observed that only in the case of very elaborated worked examples that were available during the problem-solving, some transfer could be made from the worked examples to equivalent test problems. It seemed that most students heavily relied on a syntactical approach, that is, replacing the numbers in the examples by the ones in the test problem. This conclusion was supported by the finding that hardly any significant transfer could be made to test problems that were not equivalent but only similar to or a special case of the given worked example. Sweller and Cooper (1985) presented similar findings using purely algebraic problems without any realistic context. A very significant improvement of solving speed and accuracy was found, using worked examples instead of practice problems, but also here transfer problems were reported.

The effectiveness of worked examples depends on the structuring of the example. For instance, worked examples in geometry were observed to be less effective. This was hypothesized to be due to an overload of information sources such as pictures, equations, and text (Sweller & Tarmizi, 1988). Several studies using cognitive load theory confirmed this hypothesis (see, for example, Sweller, Chandler, Tierney, & Cooper, 1990).

An important remark is that in all studies discussed so far the worked examples were offered to students in the beginning phase of skill acquisition. Some very short, basic information on the principle or strategy to be learned was provided, but students did not have any experience with solving problems using the principle or strategy. Indeed, studying worked examples was found later on to be effective only in this first stage of learning a new principle. Kalyuga, Chandler, Tuovinen and Sweller (2001) claimed that for students who already have sufficient domain knowledge, solving problems is more effective than studying worked examples.

Three main issues

Though many of the previously discussed studies reported significant positive effects of studying worked examples, also limitations and difficulties were observed. I distinguish the following problems that can arise if students learn by worked examples:

• Transfer issues: students are able to replicate the worked examples and solve equivalent problems, but the worked examples do not always provide the profound understanding that enables students to solve problems that look different or re-quire a slight adjustment of the technique. This decreases the speed advantage of studying by worked examples, since more worked examples are needed (Reed, 1985, Sweller & Cooper, 1985).

• Timing issues: studying worked examples can be ineffective, or even counterpro-ductive, if it is not done in an appropriate stage of skill acquisition. Studying worked examples seems more effective than problem solving only in the first stage

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of skill acquisition: the effectiveness of worked examples seems to decrease during the acquisition of domain knowledge (Kalyuga et al., 2001).

• Structuring issues: the beneficial effects of studying worked examples can be can-celed out, because the structuring of the worked example (for instance, the use of different information sources) is asking too much of the cognitive capacity of the student (Sweller & Tarmizi, 1988).

In the next three sections, I discuss several studies in which solutions to these problems were proposed as to maximize the positive effects of worked examples. In other words, the problems are translated into instructional principles that ought to be taken into account in order to use worked examples effectively.

2.3 Transfer issues

Making a transfer from studying a technique demonstrated in worked examples to ap-plying this technique to a different problem requires a certain amount of understanding. A student should not be just imitating a worked example by syntactically mapping, but should have derived some general principles by studying the worked examples (see, for example, Chi, Bassok, Lewis, Reimann, & Glaser, 1989). Chi, deLeeuw, Chiu and La-Vancher (1994), doing research on learning different subjects at high school level, claimed that a way to enhance this is by stimulating the student to self-explain the steps in the worked examples. In this section I elaborate on the concept of self-explanation and the ways to elicit self-explanation. Hereafter I discuss some other instructional principles that aim at solving transfer problems.

Self-explanation

Chi et al. (1989) investigated to what extent and in what way students give spon-taneous self-explanations when they are studying a worked example of a mechanics problem. They found that students who usually performed well in the subject had a different self-explanation style than students with lower achievements. ‘Good’ students gave more self-explanations and of better quality, in the sense that they related steps in the worked example to their knowledge of the subject, whereas ‘poor’ students gave less self-explanations and of lower quality, in the sense that their explanations were wrong or incomplete. Furthermore, ‘good’ students seemed to be better at detecting their own failures or misunderstandings, and were more able to use the worked examples as a ref-erence in certain steps when solving another problem. In contrast, ‘poor’ students did not know what their misunderstandings were and tended to look for a complete solution in the worked example. This implies that there is a relation between the amount and quality of self-explanations on the one hand and the ability to detect misunderstandings and to use the worked example in an efficient way on the other hand.

Not only do students with a good basic mathematical knowledge give better self-explanations, the reverse is seen as well: Renkl (1997) noticed that among students

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with equal mathematical skills, the ones who give more and better self-explanations during studying worked examples, gain more out of it. As therefore might be expected, actively stimulating self-explanation by explicitly asking students to explain each step is an effective way to enhance learning (see, for example, Chi et al., 1994). In addition, Renkl (1998) claimed that stimulation of self-explanation increases the transfer capability of the student. It is therefore important in which ways self-explanation can be elicited.

Eliciting self-explanations

Atkinson, Renkl and Merrill (2003) translated the observation that eliciting self-explanation increases learning gains and transfer capability into worked example design. They tegrated self-explanation prompts in the worked example: students were asked, for in-stance, to tick the box with the correct explanation of a certain step. This gave signif-icantly better results than worked examples without self-explanation prompts. Ticking a box with the correct explanation, that is, answering a multiple choice question about the explanation, is one way to elicit self-explanation. Other, less suggestive options are to let the student write down the explanation or to let the student fill in certain gaps in an explanation (see, for example, Renkl, 2015). As will be discussed in the next section, the appropriate amount of guiding is determined by the stage of skill acquisition of the student.

The classical idea behind self-explanation is that students elaborate on an individual example and explain each step (Chi et al., 1989). Another possibility is not to explain the steps of a single example, but to compare different examples. This comparative type of self-explanation is particularly helpful in learning to distinguish problem categories, and is therefore essential to enhance transfer capability (see Gerjets, Scheiter, & Schuh, 2008; Nokes-Malach, VanLehn, Belenky, Lichtenstein,& Cox, 2013).

Another interesting variation was brought up by Grosse and Renkl (2007) and is the deliberate inclusion of mistakes in the worked example. The idea is that a mistake works in a similar way as a self-explanation prompt: Students are encouraged to explain the steps to themselves in order to check their correctness. Grosse and Renkl found that worked examples with occasional mistakes do indeed increase learning gains, as long as the knowledge of the student on the subject is not too low. In the case of absolute beginners, worked examples without mistakes worked significantly better.

Meaningful building blocks

In order to make a transfer from an example to a different problem, students have to be able to split the steps in the worked example and be aware of the function of each step. This is needed, according to Catrambone (1998), for application of steps in a different order to suit the specific situation of the problem. This approach, where a technique is not seen as a fixed order of steps, but where each step has a function and a meaning, is called the meaningful building blocks approach (Renkl, 2015). To stimulate the student to approach the worked examples in this way, the worked example can be structured as to draw attention to the separate steps. This could be done by using, for instance,

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circles, labels or blank lines to isolate the steps. Spanjers, van Gog, & van Merri¨enboer (2012) also tested stimulating students to segment the worked example for themselves, but they observed that this induced only extra cognitive load without having significant beneficial effects.

Example sets

Another way to look at transfer issues is to consider it as the inability of the student to distinguish between different subcategories of problems, which makes it hard to select the correct solving strategy. The student clings to the strategies that are demonstrated in the worked examples but has problems recognizing when to apply which strategy and when to combine different strategies (Rittle-Johnson, Star & Durkin, 2009). A possible solution to this issue is the use of structure-emphasizing example sets: carefully selected worked examples that are presented in a certain order as to emphasize the characteristics of each problem category. A method of composing such a set is to present problems of the same type in worked examples with varying cover stories, but to use the same cover stories across different problem types (see, for example, Quilici & Mayer, 1996). This way of example set selection aims at training the recognition skills of the student, and to prevent a focus on (often misleading) surface features such as key words that would point to a certain problem category. In order to maximize this effect, the use of a structure-emphasizing example set is recommended to be combined with prompts that ask for comparative self-explanations (see for example, Scheiter et al., 2003; Richland & McDonough, 2010).

2.4 Timing issues

In this section I discuss at which moment in a learning process worked examples are seemingly most effective, and how to schedule the use of them. To answer these questions it is important to first elaborate on the knowledge-gain reversal effect concerning the relation between studying worked examples and solving problems.

The knowledge-gain reversal effect

Kalyuga et al. (2001) claimed that studying worked examples is more effective than solving practice problems in the earliest stage of skill acquisition and explained this using cognitive theory. In the earliest stage of skill acquisition, students have very little knowledge of the problem solving techniques that are specific for the domain that is studied. When trying to solve a practice problem, they therefore have to rely on other, more general strategies, for example the earlier mentioned syntactical mapping, also called a shallow copy-and-adapt strategy. Another example is the key word strategy, where a word in the cover story of a problem is used as an indication to use a certain technique, without understanding why or being certain if the technique is indeed useful. These shallow strategies that are not specific to the domain that is studied, do not contribute to gaining profound domain knowledge, but they do impose cognitive load.

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Therefore studying worked examples that provide the student with knowledge of domain-specific techniques is preferred in this stage.

In a later stage, when the student has gained advanced domain-specific knowledge, the reverse happens. Studying worked examples is now of less use because the student already knows how to use the technique. Explaining the steps in the example has become a redundant activity, and unnecessary cognitive load is imposed by it. Practicing to solve problems independently is now more useful because this supports the skill of applying the already gained knowledge. This decrease of effectiveness of studying worked examples, parallel to the increase of effectiveness of solving problems, is called the expertise-reversal effect or the knowledge-gain reversal effect (see Renkl & Atkinson, 2010, for a concise description of this effect).

Fading worked-out steps

Clearly, there is a large transition zone between the beginning stage of skill acquisition and the stage in which a student has reached sufficient domain knowledge. It is therefore necessary to facilitate the transition from the stage of only studying worked examples to the stage of only solving problems. A way to do this is by the gradual transformation from worked example to practice problem by means of fading worked-out steps. At first a student is presented with a complete worked example. Then, a new example is given where one of the steps is omitted, and the student is asked to fill in the missing step. This goes on till studying worked examples has fully transitioned to solving problems. Both backward fading (omitting the last step first) and forward fading (omitting the first step first) have been tested, yielding equally positive results (Renkl, Atkinson, & Grosse, 2004; Renkl & Atkinson, 2010).

The fading technique can be combined with the technique of eliciting self-explanations. At first, complete worked examples are given with ‘supportive’ self-explanation prompts (such as multiple choice self-explanation questions). This self-explanation help is faded just as the worked-out steps are faded, until the student has to perform and explain all steps without help (Atkinson et al., 2003). Since learning speed and domain knowledge differs per student, Salden, Aleven, Renkl and Schwonke (2009) recommended a (digital) system that adapts to individual needs, for instance by only fading a step if the student has provided the correct self-explanation of the corresponding step in an earlier worked example.

2.5 Structuring issues

In this section I discuss the research done on the structuring of worked examples. An important issue is to deal with multiple sources of information within an example, such as pictorial, graphical, mathematical and textual information.

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Multiple information sources and cognitive load

As argued before, in the beginning stage of learning a new skill studying worked exam-ples is usually more effective than solving practice problems, inducing less (unnecessary) cognitive load. This advantage however seems to cancel out in the case that the struc-ture of the example directs attention in a way that does not contribute to learning. For example, if a worked example in geometry contains a figure, text, and equations, the student has to use his or her cognitive resources to link the various sources of informa-tion. This split-attention effect reduces the effectiveness of the worked example (see, for example, Tarmizi & Sweller, 1988; Sweller, Chandler, Tierney, & Cooper, 1990). Not only in the case of combining figures and text this can occur: also when the problem and the solution are presented in such a way that the two are difficult to link, unnecessary cognitive load can be imposed. This might be the case when both the problem and the solution consist of more than one part, and it takes effort to link the corresponding parts (Sweller, 1989). See, for example, Figure 2.1.

Figure 2.1: Example of a problem and solution consisting of different parts. From “Cog-nitive technology: some procedures for facilitating learning and problem solv-ing in mathematics and science,” by J. Sweller, 1989, Journal of Educational Psychology, 81 (4), p.464.

Integration of sources and the easy-mapping principle

There are several measures that can be taken to reduce cognitive load in the case of multiple sources of information in a worked example. The key is to either integrate the different sources of information or to help the student to integrate the different sources. The latter approach is called the easy-mapping principle: the student is supported to make a mapping between the different sources of information, that is, to link correspond-ing parts (Renkl, 2015).

The integration of different sources can be done by, for example, putting equations inside a figure instead of next to it, or by inserting sub-solutions into the problem

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statement (see Figure 2.2). If a worked example with multiple sources of information is well-integrated, all positive effects can be recovered (Tarmizi & Sweller, 1988; Sweller, 1989).

As for supporting easy mapping between the different sources of information, good results have been obtained by color coding: corresponding parts of the different infor-mation sources get the same color. This can be combined with self-explanation prompts that specifically ask for elaboration on the links between the different sources. The latter can have good effects, but is also seen to be counterproductive in some cases, if students come up with wrong explanations that lead to confusion and misinterpretations of the multiple information sources (Berthold & Renkl, 2009).

Figure 2.2: An example of sub-solutions that are integrated in the problem statement. From “Cognitive technology: some procedures for facilitating learning and problem solving in mathematics and science,” by J. Sweller, 1989, Journal of Educational Psychology, 81 (4), p.465.

2.6 Summary of instructional principles

Studying worked examples can be very effective in enhancing and speeding up skill acquisition, if implemented in the right way. I conclude with a summary of research-based instructional principles that should ideally be taken into account when using worked examples.

In order to increase the capability of the student to transfer the techniques demon-strated in the worked examples to other problems, the most important tool is eliciting self-explanations. These explanations can focus on explaining steps in one example, or on comparing different examples. Other ways to enhance transfer capability, are structure-emphasizing methods such as dividing the example into meaningful building blocks or the use of example sets.

It is important to let students study worked examples in the appropriate learning stage. Studying worked examples is most effective in the beginning stage of skill acqui-sition. Later on, solving problems is more effective. To make the transition from worked examples to practice problems, the method of fading worked-out steps combined with self-explanation prompts has proven to be effective.

To prevent the elimination of the advantages of studying worked examples, it is essen-tial to structure the example in the right way, such that no unnecessary cognitive load is imposed. If multiple sources of information are used (figures, equations, text), it is

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recommended to either visually integrate the sources (e.g., placing the equations in the figure), or to offer support to the student in linking the different sources. A successful way to do this is by color coding.

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3 Research Questions, Research Design

and Methodology

In this chapter I present my research questions, link them to previous research and describe the expected results. Hereafter I discuss the design of my research and elaborate on the methodology.

3.1 Research questions and expectations

I have two main research questions. Question 1 corresponds to the part of the research in which I aim to identify the problems that are related to the intensive use of worked examples during skill acquisition and problem-solving. Question 2 corresponds to the part in which I aim to develop and propose recommendations for worked example design, driven by the problems identified.

1. Which types of errors can be identified when psychobiology students in the calculus course solve problems relying on worked examples?

2. To what extent do self-explanation prompts that emphasize meaningful building blocks in a worked example support the summer school students in using an effec-tive copy-and-adapt strategy?

I now elaborate for each question on the exact meaning and the expectations of the outcomes.

Research question 1: meaning and expected outcomes

I go from left to right through the important words in the first question.

Types of errors I am speaking of types of errors because I aim to make a classification of the observed errors.

Errors The word errors refers to mistakes or difficulties in problem solving, that are presumably the result of or related to the use of worked examples.

Problems In this question, I mean problems in calculus. Concrete examples are the computation of an integral, finding the derivative of a function or solving a differential equation or initial value problem.

Relying on Relying on ought to be understood in a somewhat broad sense: it is meant to cover the behavior of all students that make use of one ore more worked examples

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during problem solving, including both the ones that follow each step precisely, and the ones that ‘consult’ the worked example once in a while.

In order to formulate the expectations on the outcomes, I go back to the instructional principles discussed in Chapter 2, which give some hints on the consequences of solving problems relying on worked examples.

Firstly, timing principles seem to be violated when students mix the activities of studying worked examples and solving problems. From previous research it is known that it is most effective to begin with studying numerous worked examples illustrating one principle, and to start practicing problems only when sufficient knowledge of the principle has been gained (Kalyuga et al., 2001). In the case of psychobiology students who rely heavily on worked examples, it seems that either this transition is made really quick (e.g, in one or two exercise classes of 2 hours), or the problem solving phase is entered ‘too early’. Based on the studies on the knowledge gain reversal effect, it is expected that students that enter the problem solving phase before gaining enough knowledge on the principle to be learned, resort to copy-and-adapt strategies and focus on shallow features of the problem and the example (Renkl & Atkinson, 2010). This might result in errors that point at random copying behavior.

We can also connect these strategies to transfer principles: when a student focuses on shallow features of the example and the problem, the ability to provide correct and thorough self-explanations decreases, and therefore transfer problems can be expected (Renkl, 1998). A typical error in which transfer problems could become clear is for instance the inappropriate application of a step from the example in a problem that requires a different action.

As for structuring principles, a possible expectation based on cognitive load theory is that extra cognitive load will be induced by switching between the worked example and the problem, in the same way as the imposing of cognitive load by switching between dif-ferent sources of information within one worked example (e.g, Tarmizi & Sweller, 1988). This would decrease effectiveness of using worked examples. Another approach is that some students might actually be using the worked examples and the problem to actively practice comparative self-explanation (between the problem and the examples), in which case the activity would be beneficial for gaining profound knowledge of the principle to be learned and the problem types to which it can be applied (e.g, Gerjets et al., 2008). I expect both cases to occur, and therefore I also expect to observe difficulties related to cognitive overload: a typical error I expect to see is a complete lack of self-explanation.

The expected outcomes can be summarized as follows. When students solve problems relying too much on worked examples, important timing principles of learning by worked examples are violated. I expect this to lead to the use of shallow strategies and corre-sponding errors that show copy-and-adapt behavior. Furthermore I expect to encounter difficulties in providing complete self-explanations and errors caused by inappropriate transfer from the example to the problem. Lastly I expect that cognitive load imposed by linking the example to the problem will sometimes lead to a complete lack of

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self-explanations and problems to come up with a strategy to solve the problem.

Research question 2: meaning and expected outcomes

I go from left to right through the important words in the second question.

Self-explanation prompts I mean self-explanation prompts within the worked example: empty boxes in which the student is asked to write down his or her thoughts on or explanation of the worked out step below the box.

Emphasize meaningful building blocks The self-explanation boxes emphasize the struc-ture of the worked example by sectioning into meaningful building blocks: the boxes separate the solution steps.

Effective copy-and-adapt strategy A strategy of using a worked example to solve a similar problem by copying some parts and adapting other parts. The strategy can be seen as effective if the student justifies which parts are copied, and which parts are adapted. Also the student has to be aware of the differences between the worked example and the problem, and adapts the worked example steps, or adds steps, according to these differences.

The reasons to choose a combination of self-explanation prompts and meaningful building blocks will be discussed in detail in Section 7.1. After all, the research in the summer school is done in order to explore possible solutions to the problems identified in the psychobiology group. Therefore the choice of self-explanation prompts and meaningful building blocks follows from the conclusions of the first part of the research.

Both the use of self-explanation prompts (see, for example, Renkl, 1998) and the use of meaningful building blocks (see, for example, Catrambone, 1998) are techniques that are aimed at increasing the transfer capability of students. Therefore I expect to find that students will be supported by the self-explanation prompts that emphasize meaningful building blocks in transferring the knowledge derived from the worked example, to the problem solution.

Concretely, I expect that the students will give problem solutions that correspond to the explanations they provided in the self-explanation boxes, which would point at conscious and consistent copy-and-adapt behavior. I also expect that most student will be able to provide self-explanations of the steps, supported by the division into meaningful building blocks. I furthermore expect that the meaningful building blocks will draw extra attention to the differences between the worked example and the problem, and students will (attempt to) adjust the problem solution accordingly.

I expect that most errors that will be made are ‘normal’, mathematical errors, rather than errors related to the reliance on a worked example. However, I hope to also observe some inconsistencies between problem solutions and self-explanations, providing insight in the limitations of the use of self-explanations and meaningful building blocks.

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The expected outcomes can be summarized as follows: I expect to find that self-explanation prompts that emphasize meaningful building blocks support the summer school students in using an effective copy-and-adapt strategy. I expect the example-related errors that are still made to provide insight in the limitations of these forms of support.

3.2 Research design and methodology

I split up my research into three parts.

The first part was the exploratory phase of the research, in which I made observations in the exercise classes of the psychobiology course. This phase took eight weeks, with weekly exercise classes of two hours. During the classes, I observed and recorded con-versations between students and teacher and took small interviews with students that seemed to have difficulties with the material. The goal of this first phase was to gain insight in the behavior and difficulties of the psychobiology students and to use this to formulate research questions.

The second part consisted of one-to-one thinking aloud sessions with a selection of psychobiology students, in which the student was thinking aloud while solving a problem with the help of worked examples. This was aimed to provide answers to the first research question.

The third part involved an experiment in the summer school group, in which the use of self-explanation prompts that emphasize meaningful building blocks was tested. This was done in order to answer the second research question.

I now go into detail on the methodology of the one-to-one thinking aloud sessions with the psychobiology students and the experiment in the summer school group.

Methodology of the thinking aloud sessions

After the observations in the exercise classes that led to my research questions, I spoke to nine students of the psychobiology group in one-to-one sessions. In these sessions I asked the student to solve a problem using one or more of the worked examples available in the course material. Students were selected on the basis of their difficulties with the course. I only selected students who said to find the course hard, because I aimed to observe errors. More information on the students in the one-to-one sessions will be given in Chapter 4. I chose to let the students think aloud during the sessions because I wished to capture self-explanations. I was interested in the way the student explained the worked example steps and in how this related to the errors that were made. The sessions were build up as follows:

1. The student chose a subject that he or she wished to practice.

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3. I asked the student to select one or more worked examples in the digital learning environment SOWISO that he or she considered as possibly helpful in solving the exercise.

4. I asked the student to try and solve the problem, using the worked example(s). I told the student that if needed, he or she was allowed to use other sources such as lecture notes or any information in SOWISO.

5. The student tried and solved the problem, while thinking aloud. Our conversations were audio-recorded and I collected the student’s notes for solving the problem.

Sometimes I interrupted or responded to the student, always with one of the following reasons in mind: To

• ask for clarification of what the student was saying, or;

• encourage the student to think aloud, or;

• ask why the student took a certain incorrect step, or;

• give some information when essential (pre-)knowledge was missing.

In the first two cases, the interruption was purely made to make sure that the recordings would be understandable and informative later on. The third reason for interrupting can be justified by recalling that I wished to gain as much information as possible on the way the students used the worked examples and on the self-explanations that the students gave. Possibly, the interruptions were also helpful for the students in correcting their mistakes, forcing them to think about the explanation of a certain step. However, interruptions were only made after the student already gave an incorrect explanation or made an incorrect step, so this did not prevent the identification of errors.

In the last case, I made an interruption in order to make sure that the student was not hindered by missing pre-knowledge and could continue trying to solve the problem. This kind of interruption could for instance be necessary if a student working on substitution would know that he or she has to differentiate in a certain step, but would not know the derivative of sin x. This knowledge is not relevant for learning the principle of substitution, and therefore it would be needed to tell the student that the derivative is cos x, to make sure the student can continue the problem solution, making errors that are related to the use of the worked example.

Methodology of the summer school intervention

After identifying error types related to the reliance on worked examples in the psychobi-ology group, I conducted an experiment in the summer school group. Based on the error types identified, two methods of worked example design described in previous research were selected: self-explanation prompts and meaningful building blocks. In the summer school experiment, I studied how the combinations of these forms of support worked in

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the case of solving a problem relying on a worked example.

I selected the subject of differentiation, and in particular maximization and minimiza-tion, because this subject seemed suited for designing problems that required the appli-cation of multiple steps. Also, lots of variation could be incorporated in these problems, without requiring substantially different knowledge. For instance, both maximizing and minimizing require solving f0(x) = 0, but the problem statement and conclusion are dif-ferent. This is similar to the variations in the integration problems in the thinking aloud sessions with the psychobiology students, and therefore similar example-related errors could be expected. For this reason, the subject of maximization/minimization seemed appropriate for observing if students are able, with the provided forms of support, to deal with differences between the problem and the worked example.

The instructional material used for this experiment can be found in Appendix B and consists of a worked example and a problem to be solved. In the solution of the worked example, self-explanation boxes are inserted, placed in such a way that the solution is sectioned into meaningful building blocks. Below, the problem statement of the worked example and the problem to be solved are given.

The problem statement of the worked example Two functions f and g on the segment [−1₂,5₂] are given by:

f (x) = x3− 3x2_{+ 2x}

g(x) = −1 3x

The graphs of f and g are given in the figure. For which x between 1 and 5₂ is the vertical distance between f and g minimal?

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The problem statement of the problem to be solved We look at two functions f and g:

f (x) = 2x g(x) = 2√x

The graphs of f and g are given in the figure. What is the maximal vertical distance between f and g on [0, 1]?

Next to some subtle variations in formulation, there are a few differences between the worked example and the problem:

• In the worked example, the upper function is called f and the lower function is called g. In the problem, this is the other way around.

• The function types are different: in the worked example, both functions are poly-nomials. In the problem, the second function contains a square root, and therefore the equation is slightly different to solve.

• In the worked example the student is asked to compute the x-value. In the problem the y-value is asked.

• The worked example requires the computation of a minimum, whereas the problem is about a maximum.

The experiment was conducted at the moment in the course that differentiation was already dealt with. Also the computation of extreme values had been discussed. How-ever, the particular case of maximizing or minimizing the distance between two graphs was new to all students. In other words, the students had all necessary pre-knowledge but were also still in the beginning phase of acquisition of differentiation skills. The experiment was conducted as follows:

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2. The students were instructed to first study the worked example and to fill in the self-explanation boxes. They were asked to provide, if possible, a self-explanation for the step below the box, and to also write down how confident they were about their explanation. If the student was not able to provide an explanation, he or she was instructed to write down as specifically as possible which parts of the step were unclear.

3. The students were asked to try and solve the problem after he or she completed the boxes in the worked example, and to explicitly write down all of their steps. They were allowed to use as much time as needed.

4. The students handed in all material after they completed (or were completely stuck in) the problem, and wrote their name on it, to enable me to ask for clarification afterwards.

The reason for asking as much explanation as possible, both in the worked example boxes and in the problem solution, was to collect as much information as possible on the link between the self-explanations (or self-explanation problems) and the (errors in the) problem solution. This was aimed at giving insight in the effectiveness of the copy-and-adapt strategy of the student.

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4 Research Setting

In this chapter I describe the educational setting of the research. I divide this part into a description of the setting of the psychobiology group and a description of the setting of the summer school group.

4.1 Educational setting: the psychobiology group

In this section I go into the educational details of the psychobiology group and the calculus course they took. I first describe the student profile, then elaborate on the set-up of the course and discuss the use of the digital learning environment SOWISO. I conclude with a description of the worked examples used in the course, and evaluate them in terms of the design principles discussed in Chapter 2.

The student population

During the research the psychobiology group was in the second semester of the bachelor program at the University of Amsterdam. The mean age of the students was around 19 years. The course they took is called basiswiskunde in de psychobiologie (basic math-ematics in psychobiology). Students had a background of high school mathmath-ematics A (focus on statistics and word problems, 25% of the students) or mathematics B (focus on calculus and geometry, 75% of the students). For students with a background of mathematics A, the topic of integration and the topic of trigonometric functions were completely new. The topics of complex numbers, differential equations, and Fourier analysis were new to all students.

For the one-to-one thinking aloud sessions I selected nine students that, according to themselves, found the course material difficult and needed some support. In Chapter 5 and 6, the analysis of the thinking aloud sessions will be presented, for which I will use excerpts from the transcripts. In Table 4.1 some background and results are given of the four students that occur repeatedly in the excerpts. These students I denote by S1, S2, S3 and S4. Two of them have a background in mathematics B, the other two in

mathematics A.

Course set-up and digital learning environment SOWISO

The course basiswiskunde in de psychobiologie took place during ten weeks (February -April, 2015). The following topics were dealt with:

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Age HS maths Grade HS maths Grade exam BWP Grade retake BWP

S1 18 Mathematics B 6 7

-S2 19 Mathematics B 8 4.5 8.5

S3 20 Mathematics A 8 4 6

S4 18 Mathematics A 8 5 7

Table 4.1: Background information of the four students, denoted by S1 till S4, used to

illustrate the analysis in Chapter 5 and 6. Grades are on a scale from 1 to 10. High school is abbreviated as HS, and the course basiswiskunde in de psychobiologie is abbreviated as BWP.

• Chemical computation

• Functions

• Differentiation

• Differentials and integration

• Complex numbers

• Fourier analysis

• Exponential growth

• Exponentially limited growth

• Logistic growth

• Differential equations

The course consisted of lectures (2 hours per week), exercise classes (2 hours per week) and self-study activities in a digital learning environment, called SOWISO. Students made compulsory weekly tests, during which they were allowed to use all of the course material. The ones coming to the exercise classes, usually made the tests there, with help of the teaching assistants. All exercises, instruction texts, tests and other course material, including worked examples, were available in the digital learning environment SOWISO. See Figure 4.1 for an impression of the course page. No books or other written course materials were used.

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Figure 4.1: Course page of basiswiskunde in de psychobiologie in digital learning envi-ronment SOWISO.

Worked examples in SOWISO

For every topic, several randomized worked examples were given in SOWISO, next to theory. I use the section on integration techniques to illustrate the role of worked exam-ples within the digital environment. I choose this section to analyze, because this was the most frequently chosen subject among the students in the thinking aloud sessions. I evaluate the design of the worked examples in terms of the instructional principles recommended in previous research. This I do in order to get some idea of the factors inherent to the example design that could influence learning gains, next to the influence of the way students use the worked examples.

A typical worked example in SOWISO is shown in Figure 4.2. For the section on integra-tion techniques, I go through the three types of instrucintegra-tional principles as distinguished in the theoretical framework in Chapter 2.

Transfer principles: No self-explanation prompts are used. There is some division into meaningful building blocks, for instance by white space between steps. Worked examples belonging to the same problem type (e.g, substitution or integration by parts) are designed as to cover different subcategories of the problem type, such as integrals with and without limits. This corresponds with the example set principle. About four worked examples per problem type are given.

Timing principles: As for the design of the worked examples, timing principles are taken into account by (a very slight) fading of worked out steps, in the sense that steps are a little less explicated in later examples than in the first examples of a problem type

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Figure 4.2: A worked example of substitution.

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(compare for instance Figure 4.2 and Figure 4.3). No accompanying self-explanation prompts are used.

Structuring principles: Worked examples are usually presented without cover story, figures or other sources of information. No support in integrating sources is therefore needed, nor used. Extra cognitive load might be imposed in some cases by lots of (explanatory) text or the presentation of multiple methods or notations for one solution (see, for example, Figure 4.2).

4.2 Educational setting: the summer school group

In this section I discuss the educational details of the summer school group and the course they took. I shortly describe the student profile and then describe the set-up of the course.

The student population

The summer school group consisted of 14 students taking a course in basic calculus and statistics, as a prerequisite for starting a bachelor’s or pre-master’s program in economics or finance (13 students) or psychology (1 student). Most students had a background of mathematics in Dutch secondary education on HAVO wiskunde A level (higher general education, 12 students) and others on MAVO (lower general education, 2 students). In practice, the level of mathematical pre-knowledge varied widely. Also, for all students it had been more than a year since they took mathematics classes.

Course set-up

The summer course in basic calculus and statistics (‘wiskunde A’) took place in July 2015 at the University of Amsterdam. It consisted of four weeks of lectures (4 x 2 hours per week) and exercise classes (4 x 2 hours per week), dealing with the following topics:

• Basic algebra: fractions, roots, powers

• Functions and equations: linear functions, quadratic and power functions, root functions, broken linear functions

• Exponential functions and logarithms

• Differentiation and application: optimization, increasing and decreasing functions, inflection points

• Basic probability theory: combinatorics, normal distribution, binomial distribu-tion, density functions

The students worked with a book, basisboek wiskunde by Jan van de Craats and Rob Bosch. Next to this, students used a syllabus that I designed for this course, containing

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multiple worked examples for every topic. Just as in the psychobiology course, some topics, for example differentiation, were completely new to the students. The experi-ment took place in the third week when differentiation and optimization was already dealt with in the last two lectures. In practice, this means that many students had dif-ferentiated f (x) = x2 for the first time two days before the experiment, and computed their first maximum the day before. This rapid succession of learning new techniques was characteristic of both the psychobiology course and the summer school course. This time pressure, forcing students to start practicing problems right in the beginning phase of skill acquisition, is an important feature of the particular setting in which I study the use of worked examples.

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5 Analysis Framework

In this chapter I discuss the framework for the data analysis of the thinking aloud sessions with the psychobiology students and for the data analysis of the summer school experiment. I illustrate both frameworks by some examples.

5.1 Analysis framework of the thinking aloud sessions

I start with the analysis framework of the thinking aloud sessions with the psychobiology students.

Analysis framework

To analyze the data of the thinking aloud sessions, I used methods from grounded theory, based on the descriptions of Teppo (2015) and Cohen, Manion, & Morrison (2007). The reason for choosing grounded theory as the analysis framework was that no classifica-tion of example-related mathematical errors could be found in previous literature, and therefore the classification had to be purely grounded in the data.

The data analysis started when about half of the data were collected, by listening once to all recordings, denoting all errors that were made. For all errors, I considered and noted down if it could be related in some way to the reliance on worked exam-ples. For this, I investigated whether the student provided any self-explanation of the relevant worked example steps, and how this self-explanation related to the observed error. All data that were collected afterwards, were treated in the same way, where all newly observed were compared to the errors found before. This process led to a rough classification of errors.

I made a selection of the data to transcribe, where I decided to transcribe all conver-sations with students that made at least one error that seemed related to the reliance on a worked example. By constant comparison of the errors observed to the classification, making modifications to the classification if an error was observed that was not accounted for yet, a final classification fitting all data was developed. The final classifaction will be given in Section 6.1 (p.44).

In the next subsection I give an excerpt of a conversation with a student in order to show how the analysis was done and how errors were identified and interpreted.

Illustration of the analysis

I discuss an excerpt of a thinking aloud session with student S2, who was working on

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are of different types, as will become clear in the next chapter. The excerpt shows that different errors can be interrelated and enables me to demonstrate how I separated dif-ferent errors and how I decided if the error is related to the reliance on worked examples.

I go through all of the errors that can be identified, discussing for each error the following three questions:

1. Which error is made?

2. Does the student provide a self-explanation for the relevant step in the worked example?

3. Is the error related to the reliance on the worked example?

In the answer to the first question I make clear what I consider to be the error. The second question is added in order to get some idea of the nature of the error that is made. The way in which the students explains the example step is key to understanding if the error is related to the reliance on the worked example and therefore essential to answer the third question. The third question is important since I only wish to classify the errors that are related to the use of worked examples.

Student S2 is trying to solve an integral by partial integration, using a worked

exam-ple. In Table 5.1 the relevant parts of the worked example and the correct problem solution are given. Below I show an excerpt from the conversation I had with student S2 when she started solving the problem. S2 indicates the words of the student, and R

indicates my words (the words of the researcher). For the full transcript, see Appendix A.

S2 integration by parts

S2 All right, so we set v = ln(x) and u = x3 and we get the integral of

u · dv, that is equal to uv minus the integral of v · du and that is equal to x3 times ln(x) minus the integral of v, so that’s ln(x)..

R Cause how do you see that you have to do x3 times ln(x)? S2 There’s uv here and I think that they fill in...

R Okay.

S2 Except that there is a minus before.. Eeeer.. then it might be the

derivative.

In this part I identify three errors, which I denote by E1, E2 and E3. I discuss them one

by one.

E1: Using v instead of dv

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Worked example Problem

R x · e−x _dx R2

1

x3· ln(x) dx

1) u = x and dv = e−x u = ln(x) and dv = x3

2) du = dx and v = −e−x du = 1_xdx and v = 1₄x4

3) R x · e−x_{dx =}_{R u dv} R2 1 x3· ln(x) dy = 2 R 1 u dv 4) = uv −R v du =huvi2 1 − 2 R 1 v du 5) = −x · e−x−R −e−x_dx h_{ln(x) ·} 1 4x4 i2 1− 2 R 1 1 4x4· 1 xdx 6) .... ....

Table 5.1: The relevant parts of the worked example (left) and the correct solution to the problem (right).

In the particular worked example that S2uses, but also in the rest of the course, u

and dv are used to denote the two parts in partial integration. Student S2 instead

uses u and v, without adjusting the formula used for partial integration, and as a result she initially fills in x3· ln(x), where she should have integrated her v first.

No, student S2 does not reflect on the notation in the worked example or give a

reason to choose her own notation in this way.

Student S2 seems to just make a reading or writing error, or she might use a

notation that she has been used to at secondary school. She seems unaware that her notation is different from the one in the worked example. The occurrence of the error is therefore seen as unrelated to the reliance on the worked example. Actually, it is the worked example that makes her aware of her mistake later on.

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E2: An inappropriate choice of u and dv

Student S2 chooses ln(x) as dv and x3 as u. This is the unwise way of choosing

them: the integral function of ln(x) is complicated, and the derivative of x3 does not make things easier.

No, student S2 does not explain why in the worked example, x is chosen as u and

e−x as dv. It seems that S2 does not think about the possibility to choose them

the other way around, she automatically chooses the left function as u and the right function as dv. Her implicit self-explanation seems to be that this is ‘some standard rule’ or just ’always like this’.

Yes, definitely: the way to choose u and dv is taken over from the worked example ‘without thinking’.

E3: An incorrect interpretation of −e−x

After finding out that −x·e−xis not exactly equal to u times dv (in her notation, u times v), since there is a minus sign, student S2 suggests that dv = e−xmight have

been differentiated. The interpretation is incorrect, though in the special case of this worked example the confusion is understandable: the derivative of e−x equals the anti-derivative.

Yes: in this case, the error is in the self-explanation of the step in the worked example. The error did not result in any wrong computation, because S2corrected

her error in time (see the full transcript in Appendix A).

Yes, definitely. The error does not even involve any mathematical mistake: student S2 knows the correct derivative of e−x and uses this knowledge. The reason for

this error is that she is looking at a special case in which the derivative happens to be equal to the anti-derivative. In other words, this error is fully related to the reliance on the worked example.

Setting the Right Example