4

Worked examples and instructional design

Before going on, try Activity 4.1.

I once attempted to learn the rudiments of the programming language Prolog from textbooks. At one point I found myself reading a section that attempted to explain the concept of recursion and presented about half a dozen examples. As I read the section I felt that the concept was quite well explained and the examples seemed readily understandable. At the end of the section there were some exercise problems and so, with some confidence, I got a sheet of paper and had a go at the first one. I quickly discovered that I hadn’t a clue where to begin to solve the problem.

Now a couple of reasons might be adduced for this state of affairs. My first – fortunately fleeting – thought was that I just couldn’t do Prolog; I wasn’t cut out to be a computer programmer. (This thought could lead to what Bransford and Stein [1993] called a “mental escape”: when presented with a problem from a particular domain the solver exclaims “Oh no, I can’t do algebra” or geometry or statistics or whatever. The solver gives up before even reading the problem because of a kind of learned helplessness.)

My second thought was that I had simply missed something when I had read the section. So I went back over the worked examples looking for one that seemed similar to the exercise problem I was trying to solve. I couldn’t find one. I still had no idea how to solve the very first exercise problem, so I did what most people probably do in such circumstances – I looked up the answer at the back of the book. This was revealing, not only because it told me the answer, but also because it showed me that the section I had just read had not, in fact, told me how to solve this particular exercise problem. We tend to take it on faith that the textbook writer has provided all the information we need to solve exercise problems, and if we can’t manage it then it must be our “fault”. This may well not be the case for a number of reasons (indeed Jonassen [1997] has argued that “worked examples should not be developed using experts” [p. 76]). If I had gained a deeper understanding of the concept of recursion and how it manifests itself in Prolog from reading the section, then perhaps I could have solved it, and presumably the author had believed that he had provided enough of an explanation to produce just such a deep understanding. Nevertheless, I would venture to suggest that students reading textbooks in domains they are unfamiliar with do not always understand things particularly well on a first reading.

One way of examining the difficulties facing students engaged in textbook problem solving is by looking at the processes involved in using worked-out examples to solve problems. There is an important distinction between the kinds of analogical problem solving discussed previously and the use of an example as an analogy in a textbook. So far, we have concentrated mainly on analogy as the transfer or mapping of knowledge from a familiar domain onto a less familiar one (far transfer). In textbook examples, on the other hand, the student is trying to use an example from an unfamiliar domain to solve a problem in the same unfamiliar domain (near transfer).

Analogical reasoning works when you can reason from a domain that you understand well to solve a present problem that is puzzling. In textbook problem solving, the example and the exercise problem are both in the same domain and the student, who is presumably a novice, does not yet understand the domain, otherwise the student would not be a novice. This, in a nutshell, is what makes much problem solving from textbook examples difficult.

Difficulties facing textbook writers

Textbooks writers face a number of difficulties when writing textbooks. To make the task a little easier they have to make some assumptions about the reader.

Assumed prior knowledge

If a textbook is aimed at a readership that has presumably reached a certain level of competence in a domain (for example, by passing exams), then the writer has a good idea of the prior knowledge of the readers. If the textbook is aimed at a more general readership, then there are likely to be parts that are better understood or better known by some readers than by others.

The Lisp textbook by Winston and Horn (1989), for example, presents an example of a recursive function in Lisp that computes the Fibonacci series (p. 73). If you already know what the Fibonacci series is, then you may have little problem understanding what the function is trying to do. If you don’t, then fortunately the textbook spends half a page explaining what it is. Having to explain what the problem statement means before explaining what the solution procedure is can present an added level of difficulty for the reader.

Assumed recall of already presented information

The writer has to make assumptions about how much the reader remembers from previous chapters. When a problem is presented in chapter 5, say, it would probably be foolish to suppose that everything presented in chapters 1 to 4 will be readily recalled. In a study into novice programmers, one of Anderson, Farrell and Sauers’s (1984) subjects had to be reminded of an earlier example from a previous chapter (problems 2–5 in chapter 21 of the Winston and Horn textbook) before she could go on and solve the problem she was working on.

Assumed understanding

Another assumption that it would be unwise to make is that all the material from earlier chapters or the current one has been understood. Several studies have shown that learners do not always have a clear idea of how much they understand from a textbook (Chi, de Leeuw, Chiu, & LaVancher, 1994; Ferguson-Hessler & de Jong, 1990; VanLehn, Jones, & Chi, 1992). In analysing the study processes of students studying physics texts, Ferguson-Hessler and de Jong, for example, found that “poor” students said “everything is clear” three times more often than “good” students, whereas their performance showed that this was not the case. Similarly, Kwon and Jonassen (2011) found that students who had low prior knowledge of the topic (computer programming in this case) tended to maintain faulty mental models despite failing to solve problems. Kintsch (1986) gives the example of trying to understand a computer manual:

The finding that poorer students felt that everything was clear more often than better students can be explained in terms of Hiebert and Lefevre’s (1986) distinction between primary and reflective understanding (see also Marton & Säljö, 1976). Primary understanding occurs when the student understands a new domain at a surface level; that is, at the same level of abstractness as, or at a less abstract level than, the information being presented. This type of understanding is highly context specific. The examples presented seem to be clear but the student is unlikely to see how they can be adapted or applied to another problem. This leads to the “illusion of understanding”, where students’ assessment of their own competency or comprehension does not match their actual performance (Dunning, Heath, & Suls, 2004; Glenberg, Wilkinson, & Epstein, 1982). Reflective understanding is at a more abstract level when students recognise the deeper structural features of problems and can relate them to previous knowledge. However, much depends on the characteristics of the learning environment, of the domain and of students themselves (Baeten, Kyndt, Struyven, & Dochy, 2010).

Assumed schematic knowledge of the structure of teaching texts

Writers need to ensure that they do not violate the student’s schema for what the layout of a scientific textbook should look like. Students are likely to have expectations about how textbooks are structured in formal domains such as mathematics, science and computer programming since they tend to have a particular stereotypical layout (Beck & McKeown, 1989; Kieras, 1985; Sweller & Cooper, 1985). With experience of such textbooks, students come to develop a schema for that type of text. Such a schema includes the default assumptions that solutions follow statements of the problem rather than vice versa, and that a particular section of a textbook will give them enough information to solve exercise problems at the end of that section. However, it may be the case that textbooks are not structured that way (see Britton, Van Dusen, Gulgoz, & Glynn, 1989).

Assumptions about generalisability

Another difficulty facing writers is whether to present close variants of the problem type or a range of variants (see Figure 4.1). Principles, concepts, how to generate an equation and so forth can be understood better by presenting a concrete example (such as Example 1 in Figure 4.1). This concrete example often acts as an exemplar or paradigm representing the problem type. However, it may be hard for the reader to recognise whether a concept, principle or solution procedure is relevant or applicable based on one example alone. Often, therefore, a range of examples is presented. If this range of examples is composed mainly of close variants of the exemplar, then the reader might be better able to abstract out the commonalities between them and hence be better able to understand the concept, principle or solution procedure and to automate the procedure for solving a subset of such problems (Guo, Pang, Yang, & Ding, 2012; Paas & Van Marriënboer, 1994). This, however, could be at the detriment of demonstrating the range of applicability of the concept or procedure and so forth (Cooper & Sweller, 1987).

If students are expected to solve distant variants of a problem, writers have to provide explicit information about the relationship between source examples and other problems of the same type (Conway & Kahney, 1987; Reed, Ernst, & Banerji, 1974). Since examples provide information about a category of problems, the more information about the features of that category which are given to the reader the better, ideally using some the form of conceptual instruction (Fyfe, DeCaro, & Rittle-Johnson, 2014).

Cognitive load

Some topics are intrinsically difficult. For example, for many people some topics such as statistics or tax law are not easy subjects to understand. If one looks at some websites that attempt to explain statistical concepts you will find that you need a good grasp of statistics to understand what they are talking about. In these cases, the way the topic is presented is difficult to understand. Understanding statistics induces an intrinsic cognitive load (see e.g., Paas, 1992) – the amount of mental effort imposed on working memory (Chandler & Sweller, 1991). If the way statistics is being explained is itself difficult to understand, then the explanation also imposes a further extraneous cognitive load. Finally, there is the load imposed by how we process the information in order to learn about the topic – that is, how we induce schemas about it. This is germane cognitive load. To sum up: extraneous cognitive load – bad; germane cognitive load – good. The trick is to design instruction that keeps extraneous cognitive load to a minimum and enhances learning (Ginns, 2006; Mayer & Moreno, 2003; Paas, 1992).

Cognitive load is a theory of instructional design based on what we know about human cognition. There has been a strong tradition of looking at example-based problem solving and learning using this approach in Europe and Australia in particular (Pierce, Duncan, Gholson, Ray, 1993; Renkl & Atkinson, 2003; Sweller, 1988; Sweller, van Merrienboer, & Paas, 1998; van Gog & Rummel, 2010; Ward & Sweller, 1990). Further developments of this theory are discussed later in the chapter.

The role of examples in textbooks

Why bother with examples at all? Isn’t a textual explanation sufficient? If people are asked to perform a complex procedure, then the best method of teaching it is by demonstration. Once people have read the text in a subject such as physics, mathematics or computer programming, they tend not to re-read it if they can help it. Instead they concentrate on worked-out examples if they are trying to solve exercise problems at the end of a section. Ross (1989a) has referred to examples as “potent teachers”, and there is a lot of evidence suggesting that they are more important for problem solving than the rest of the text. This phenomenon has been known and commented on for several decades (Anderson et al., 1984; Jonsson, Norqvist, Liljekvist, & Lithner, 2014; Lithner, 2003, 2008; Øystein, 2011; Pirolli, 1991; Pirolli & Anderson, 1985; Reed & Bolstad, 1991; Ross, Perkins, & Tenpenny, 1990; Ward & Sweller, 1990). For example, Pirolli (1991, p. 209) states: “When a learner is faced with novel goals, the preferred method of problem solving involves the use of example solutions as analogies for the target solution.” VanLehn (1990, p. 22) goes further: “Examples, exercises and other concrete examples of problem solving are the most salient parts of instruction. The verbal and textual explanations that often accompany such concrete episodes of problem solving have a secondary, indirect effect on learning.” VanLehn (1986) has referred to a “folk model” of how students learn from textbook examples and explanations. The explanations are seen as the important aspect of instruction and are assumed to be adequate for students to learn procedures for solving problems. In contrast, he argues that explanations serve as guides to help students make more accurate inductions from examples.

However, while the examples may demonstrate the procedures or algorithms for solving problems and instantiate important concepts, their use does not necessarily mean that the student has a deep understanding of, say, the mathematics involved (Øystein, 2011). That said, even experts will tend to use well-learned procedures (essentially reproductive thinking) since this is an efficient way of using their schema-based and stereotypical knowledge (Bilalic´, McLeod, & Gobet, 2009; Saariluoma, 1990). Information Box 4.1 contains some more specific examples.

According to Ross (1989b), being reminded of an earlier example can have four possible effects which amount to different roles played by worked-out examples. First, it may allow the learner to remember the details of a solution procedure rather than an abstract principle or rule (such as an equation). “The memory for what was done last time is highly interconnected and redundant, allowing the learner to piece it together without remembering separately each part and its position in the sequence” (p. 439).

Second, even if the learner can remember the rule or principle that is supposed to be applied to a problem, the learner may not know how to apply it. Activity 4.2 gives some indication of what this means.

For novices in algebra, being told the principle (the relevant equation) underlying the problem may be of no use since they do not necessarily know how to instantiate the variables. The novices still have to make a number of inferences based on domain knowledge before they can solve the problem.

Third, novices may not understand the concepts embodied in the rule or principle or may have misinterpreted them. For example, the equation in Activity 4.2 is based on the more general equation Distance = Rate × Time. Since both cars travel the same distance then Distance_CarA = Distance_CarB; and since the distances are equal, the Rate × Time for both cars must be equal, too – hence the form of the equation. Now if you know something about algebra or mathematics in general then that explanation might make sense and you can understand where the equation comes from. If you have little knowledge of mathematics, then the origin of the equation may be rather obscure. That is, you may not understand the concepts involved in the equation.

Fourth, trying to solve a current problem may force novices to extract more information from an earlier problem than they did at the time. If you saw how to solve the Fortress problem based on the Radiation problem you may have been able to abstract out information from the Radiation problem that was more relevant to “divide and converge” problems.

Principle cueing

Ross (1984, 1987, 1989b) discusses two possible scenarios in APS: the principle-cueing view and the example-analogy view. In the principle-cueing view, learners may be reminded of an earlier example by some feature or combination of features of the current one. This reminding triggers or cues the abstract information or principle involved in the earlier problem which is relevant to the current one. In the case of Gick and Holyoak’s (1980, 1983) work, the principle here would be the divide and converge solution schema; in algebra problems it might be an equation such as Distance = Rate × Time. The principle thus accessed can then be used to make sense of a new situation or solve a new problem with the same structure. The role of the surface features of problems is to cue or access a possible relevant source problem.

Holland, Holyoak, Nisbett and Thagard (1986) refer to analogues as having an “implicit” schema which is reconstructed during the solution process. In Figure 4.2, A represents a problem statement and B the goal state. The relation or set of relations between A and B forming a solution procedure is represented by the line linking them. If the problem is an instance of a category of problems, then the solution procedure used to get from A to B can be applied to other problems of the same type. There is therefore a schema implicit in the solution that can be applied to a range of problems of the same type. The schema can be made explicit to some extent by emphasising the conceptual underpinnings of the schema, for example by using a schematic picture (Chen, 2002) or by presenting concepts in advance of problem solving (Fyfe et al., 2014). The implicit schema is shown as the shaded S box in the figure.

When a source problem is accessed (A and B in Figure 4.3), then the principle underlying the solution to the source is accessed (the S on the line linking A and B) and applied to the target (C) to generate the solution (D).

In the principle-cueing view, when people are reminded of an analogy, the reminding serves to categorise the current problem. When presented with a problem involving two boats on a river going at different speeds, one might be reminded of an earlier problem (or earlier problems) of the same type and hence categorise the current target problem as a riverboat problem, or a Rate × Time problem, or whatever. The studies by Gick and Holyoak (1980, 1983) have shown that only one presentation of a problem type is required for the solver to abstract out the underlying schema. This is probably true only when the underlying schema is relatively straightforward and easily understood. More complex concepts such as recursion, say, or a particular grammatical construction in French would presumably require several examples before a schema that would help solve a new range of examples would emerge. Whether the source problem is simple or complex, the abstract principle which the source exemplified has to be already understood by the learner and the surface details in the source problem are no longer required to solve the target. The original source problem was nevertheless important in that it allowed the learner to understand the principle in question and how it is used (see also Fyfe et al., 2014).

An implication of the principle-cueing view is that solving another problem from an example involves abstracting out the principle or procedure from the example and applying it to the target. This smacks of abstraction mapping (Gentner, 1989), where an abstract principle such as an equation is mapped on to the target problem rather than the specific elements in the example. In the present case, the abstraction is “hidden” or implicit within the source example and has to be extracted before it is applied. However, the degree to which an underlying principle or schema can be abstracted out to use to solve a target is unclear.

Much of the literature on expert–novice differences has concentrated on how the correct perception of a problem can cue access to the problem schema (Bilalic´ et al., 2009; Chi, Glaser, & Rees, 1983; Larkin, 1978). This problem schema in turn suggests a straightforward, stereotypical solution method. To confuse matters somewhat, Kurtz and Loewenstein (2007) has separated out the problem schema from the solution schema as mentioned in Chapter 3. This separation was prefigured somewhat in a study by Pierce et al. (1993, p. 72), who concluded that “the quality of the base [source] schema mapped to the target and the processes involved in procedural adaptation may be relatively independent of each other.” However, a problem schema typically refers to a structured set of relationships (Gick and Holyoak, 1983). It is a declarative representation of the problem whereas the solution schema is a mainly procedural representation. Novices, however, are often unable to identify the problem schema or categorise problems accordingly. Furthermore, principle-cueing by definition presupposes that the analogiser understands the principle in the first place and how it can be instantiated in a particular problem (e.g., what values map onto the variables of an equation). For novices studying a new subject, that may not necessarily be the case.

If the solver is trying to use a complex example as an analogy in a domain that is unfamiliar, it would be unwarranted to assume that the solver has a schema, implicit or otherwise, for a problem. There may be a schema implicit in the problem but there is no guarantee that it is represented in the mind of the solver. It is likely that principle cueing is limited to either relatively simple problems where a lot of prior general knowledge can be brought to bear, or to problems where the solver already has at least a partial schema for a problem type. According to Chen (2002):

Using an example as an analogy

The view that the role of superficial features is simply to access a previous problem has been challenged by Ross in a series of experiments. According to the second view of analogising, the example-analogy view,

Much of Ross’s work was concerned with the effects of superficial similarities in problem access and use. An example of his work is given in Information Box 4.2.

Ross found that the ability of the subjects to use the relevant formula, even when the formula was given to them, still depended on the superficial similarity of the problems. The similarity between objects in the problems with the same storyline was used to instantiate the formula, so that the objects were assigned to the same variable roles as in the example. Thus, in the same/same condition (e.g., where mechanics chose the cars in the example and in the test), performance was higher than for the unrelated group. If the object correspondences were reversed – the same/reversed condition – then performance was lower than in the unrelated condition. Where it was difficult to tell which formula to use, the superficial similarity of problems with the same underlying structure led to the best performance.

When trying to make an analogy between two problems without an adequate representation of the problem structure, the usual means of instantiating variables through an understanding of what they represent is very difficult. Without that understanding novices can only rely on superficial similarities. This means that, even when learners are provided with a formula when given a test, they will still make use of an earlier example in which the principle is incorporated in order to solve the current problem. Ross’s results are therefore at odds with those one would expect from a principle-cueing view, in which the example plays no role other than as an instantiation of a schema or principle which is either already known or readily induced.

Understanding the relationship between a problem’s features and the underlying principles involved is further complicated by the degree of adaptation required of an example to solve a test problem; that is, by the degree of transfer from near to far. Helping students reach that understanding so that they can solve the problems they encounter in the future relies on the quality and nature of the instruction. Information Box 4.3 illustrates how one might go about supporting students’ understanding of the relation between the superficial features and structural, principle-based features of a problem type.

As Nokes-Malach et al.’s study showed, generating a principle, conceptual understanding or schema from examples is not necessarily an automatic process. Previous studies have shown the need for hints or remindings (Ross, 1984, 1989a). Didierjean (2003) found that participants in his experiments did not automatically generalise from a single analogue and that subjects perform better when their attention is drawn to the usefulness of generalising their knowledge during problem solving. Similarly, Goldwater and Gentner (2015, p. 151) found that conceptual understanding in the form of the ability to recognise causal patterns could be fostered by combining explanations with structural alignment (analogical comparison). They argue that this mirrors the way in which expertise develops.

The processes involved in textbook problem solving

Various representations can be derived from a textual presentation of a problem. When confronted with a word problem to solve in a textbook, a student is faced with a piece of text. The first thing the student has to do is therefore to make sense of the text itself, which requires several layers of representation (Kintsch, 1986; Nathan, Kintsch, & Young, 1992; Van Dijk & Kintsch, 1983).

First of all, there are the individual words that compose the text. Understanding these comes through our semantic knowledge of the items in our mental lexicon. From the individual words and the context of the sentence, our overall understanding of the text of a problem is constructed and so on. The initial representation of the text is a propositional representation called the textbase. Knowing what the text of a question means does not therefore entail an understanding of the problem, however.

From the textbase students have to develop a representation of the situation described in the text. This is a mental model which Van Dijk and Kintsch (1983) termed a situation model, composed of text-derived and knowledge-derived information. For problem solving to be successful, the solver has to generate all the necessary inferences in order to build a representation of the problem that is useful enough to solve it. This in turn means that novices have to have enough domain-relevant knowledge to do so.

In a later formulation of the theory, Nathan et al. (1992) divided the situation model into two. The situation model included elaborated inferences generated from an understanding of the text. Such inferences might include the fact that if two cars leave from the same point at different times and the second car overtakes the first, then both cars will have travelled the same distance at that point. The fact that both cars travelled the same distance may not be explicitly mentioned in the text.

A further representational form proposed by Kintsch and his co-workers is the problem model which includes formal knowledge, for example, about the arithmetic structure derived from the text, or the operating procedure constructed from information in the text. The ability to make inferences from texts in order to derive a useful problem model depends on the relevant prior domain knowledge of the learner (Kintsch, 1998).

The distinction between a propositional (textbase) representation of a text and the elaborated situation model was examined by Tardieu, Ehrlich, and Gyselinck (1992), who argued that novices and experts in a particular domain would not differ in the propositional representation they derived from a text but that there would be differences between the two groups in the situation model (here again the situation model and the problem model are synonymous). Tardieu et al. found that there was no difference between experts and novices on their ability to paraphrase a text (i.e., they both generated much the same textbase) but experts performed better on inference questions than novices (they had derived different situation models from the textbase).

The next section presents an example of a study where the students were unable to generate a complete situation or problem model.

Reed, Dempster, and Ettinger (1985) describe four experiments in which one example problem and solution is presented and the student is thereafter expected to solve a transfer problem, or a problem whose solution procedure was unrelated to the example (Information Box 4.4). In Reed et al.’s terminology the transfer problems were called “equivalent” or “similar”. We will look at the experiments in general and at some of the algebra word problems in particular with a view to discovering just what the solution explanations that were provided failed to explain.

Problems based on rates can be represented as a hierarchy as in Figure 4.4. Reed et al.’s study was based on the equation on the bottom left of Figure 4.4. If the source and target both involve finding the time taken to catch up with another vehicle then the mapping should be reasonably straightforward as the problems are similar in that they use the same procedure. However, if the target involves finding the speed of a vehicle when the source refers to the goal of finding the time taken, then the problem is no longer a similar one to the source. It involves an understanding at a more abstract level of the hierarchy as in Figure 4.5. “Target 4” in Figure 4.5 relates to the Target 4 problem in Table 4.3.

How difficult a problem is depends on the level of the student’s understanding of the domain and of the nature of the problem. Generating a situation model from the textbase and thence a problem model depends on the student’s prior knowledge, both factual and conceptual. So what does it mean to “understand” a problem such as these?

Understanding problems revisited

Imitative problem solving

“Understanding is arguably the most important component of problem solving (e.g., Duncker, 1945; Greeno, 1977), and representations indicate how solvers understand problems … One cannot solve a problem one does not understand (except by chance)” (Novick, 1990, p. 129, italics added). On the other hand, you can solve problems you don’t understand if they involve very little adaptation and all you are doing is copying the example (Robertson, 2000). This is imitative problem solving, where the solver, faced with a new problem, looks back to an earlier problem example in a textbook, and tries to map the surface features of the source onto the target. Imitative problem solving involves:

1. 1  Forming a representation of the target problem. Since novices are unlikely to have a “complete problem model” (Holland et al., 1986) this representation is likely to be impoverished in some way. Similarly, novices may have a poor representation of a source problem – there is no assumption that the solver fully understands the underlying solution structure.
2. 2  Mapping values across that either:
   1. a  Appear superficially to fill the same roles in both problems;
   2. b  Are perceptually or semantically similar.
3. 3  Slotting the new values into the target problem structure by trying to do the same things with the new values as was done in the source. The latter process involves inferring operators that will reduce any differences between the source and target problems in a form of means–ends analysis.

Conceptual understanding

There are several reasons why novices may fail to make elaborative inferences from a reading of a problem. First, their representations of the text are often fragmentary and incomplete, since they may not know what aspects of the text are important or relevant to the solution. Second, they require practice at solving problems, or undergo effective instructional manipulations, before they develop the necessary inference rules. In other words their declarative knowledge of the domain (or conceptual knowledge) is not necessarily in a form that can yet support inferences.

Conceptual knowledge has been referred to as “knowledge rich in relationships” (Hiebert & Lefevre, 1986). In this view, when a concept is learned it is learned with meaning by definition. Procedures, on the other hand, may or may not be learned with meaning. This leads to a paradox. If conceptual (declarative) understanding comes first, then procedures are surely also learned with meaning, since those procedures make reference to known concepts. If, however, procedures are acquired first, then conceptual understanding comes after the procedures have been learned. In other words it is possible for procedural knowledge to precede certain types of declarative knowledge.

Jonassen (2006) regards concepts as entities that change depending on how they are understood by the learner. So concepts and how they are used in, say, a procedure are intertwined and depend on the theory or mental model the learner has of concepts in use and the interrelationships of concepts. Thus concepts are not so much learned with meaning but rather the meaning can change and develop with use in context.

Approaches to the design of instruction

Cognitive load

Knowledge “rich in relationships” can often lead to what Sweller has referred to as high “element interactivity”, which refers to the complexity of the concepts or information and their interrelations (Sweller, 1994, 2010; Sweller & Chandler, 1994). Elements are those bits of information that “must be processed simultaneously in working memory to achieve understanding because they are logically related” (Chen, Kalyuga, & Sweller, 2015, p. 3). In other words, these cognitive elements refer to anything that has been or requires to be learned.

This intrinsic cognitive load is distinct from the load on working memory produced by the structure and nature of the method of instruction. Low element interactivity does not involve complex interactions of cognitive elements and so poses no great load on working memory; high element interactivity requires the learner to understand the interactions between elements rather than simply the individual elements themselves. Cognitive elements can be facts, concepts or procedures. Sweller (2010) gives learning the symbol for iron or copper as an example of a cognitive element, and learning such items would not impose much of a load on working memory and would involve low element interactivity. When several chemical elements take part in a reaction, then the learner has to understand the complex interactions of the chemical elements such as 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂. The higher level of element interactivity here produces a greater load on working memory, and this may vary depending on the prior domain knowledge of the learner.

In early formulations of Sweller’s cognitive load theory, high element interactivity was the reason for the intrinsic difficulty of to-be-learned material as it leads to a heavy load on working memory. At that time the focus of cognitive load theory was on the nature of working memory and the interaction of its subcomponents (the phonological loop, the visuospatial sketchpad, and semantic buffers; Baddeley, 2007). Since working memory is a limited capacity system, instruction should be designed to get round its limits by, for example, using multimedia (text, graphics, narrations, etc.) that rely on accessing the separate components of working memory (Gerjets, Scheiter, & Catrambone, 2004; Mayer, 2001; Mayer & Moreno, 1998, 2003; Moreno & Mayer, 2000a; Mousavi, Low, & Sweller, 1995; Sweller et al., 1998; Tindall-Ford, Chandler, & Sweller, 1997). However, more recently Sweller and co-workers have broadened out from a focus on working memory only and at the same time the boundaries between the different forms of cognitive load (intrinsic, extraneous, germane) have become fuzzier. There are ways of getting round the working memory limits by using the environment, schema-based knowledge and off-loading some of the load to long-term memory (Ericsson & Kintsch, 1995). Paas and Sweller (2012, p. 28) have stated: “The capacity and duration limits of working memory are far below the requirements of most substantive areas of human intellectual activity.” As a result Sweller and colleagues (Paas & Sweller, 2012; Sweller, 2006) have looked at the general social, cultural, evolutionary and biological contexts of learning, in which the role played by working memory is important for the acquisition of novel, culturally important (biologically secondary) information, since learning such material requires conscious effort unlike that required to learn biologically primary information (Geary, 2008).

Paas and Sweller provide a number of ways in which this view relates to aspects of instructional design. For example, learners learn more from instructional text when information is presented in a spoken form and a visual form (a graphic, image or animation) than when they are required to read the information alongside the image. This is known as the modality effect (see e.g., Moreno & Mayer, 1999; Mousavi et al., 1995; Tindall-Ford et al., 1997). Paas and Sweller (2012) argue that the apparent increase in working memory capacity may be due to a reliance on biologically primary knowledge and that “we may have evolved to listen to someone talking about an object while looking at it. We certainly have not evolved to read about an object while looking at it because reading itself requires biologically secondary knowledge” (p. 39).

With regard to the different forms of cognitive load (intrinsic, extraneous, germane), Sweller (2010) has stated that the focus has been on element interactivity as the main source of, or explanation for, levels of intrinsic cognitive load, but little has been said about the source of extraneous load. He argues in this paper that element interactivity is also the source of the load ascribed to extraneous cognitive load and hence element interactivity rather than working memory limitations has become central to cognitive load theory. What constitutes intrinsic and extraneous load depends on the goals of the learner or the instruction. Unfamiliar jargon may make learning new material difficult and so could be classed as extraneous load, but if the goal is to familiarise yourself with the jargon then it would be classed as intrinsic. Assuming a reasonably high constant level of motivation,

Under this formulation the total cognitive load is therefore determined by both intrinsic and extraneous cognitive load together, and germane cognitive load is no longer regarded as a separate, additional source of load on working memory. If either intrinsic or extraneous load increases then the overall cognitive load increases, and as a result germane cognitive load decreases as there are fewer working memory resources to devote to both intrinsic and extraneous load with a subsequent reduction in learning (see Figure 4.6). As a result, the theory in its newer form has important consequences for the development of instructional designs, although it remains the case that the goal of instructional design should be to free up as many working memory resources as possible to process new material and construct relevant schemas.

“Guided” versus “unguided” instruction

So far in this chapter we have looked mainly at within-domain transfer, particularly in relation to how examples are used in textbooks, and the limits to cognitive capacity that constrain how we learn from them. Cognitive load theory is one of a number of approaches to how we engage in learning about a new domain of knowledge that can be roughly divided into two general classes: unguided or minimally guided instruction, and guided instruction, although there is some controversy over these classifications as we shall see.

Unguided or minimally guided instruction comes under the general rubric of constructivism: a theory of how people learn by constructing meanings through interacting with the environment. As such it is a theory of learning rather than a theory of instruction, although pedagogical interventions based on this general view (which goes back to the theories of Jean Piaget) concentrate on the individual learner’s construction of meanings within their own social, cultural and experiential context: knowledge is thereby “discovered” by the individual.

Situated learning: Proponents of situated learning emphasise the degree to which learning is bound to a specific context, particularly the social context. Lave and Wenger (1991) regard learning as “legitimate peripheral participation”, by which they mean that the acquisition of knowledge and skill involves engaging in the sociocultural practices of the community of which a learner is a part. It incorporates a constructivist viewpoint in which knowledge is socially constructed (Figure 4.7). However, it need not be a specifically social context: context-dependent memory refers to the fact that we remember learned material best when the context at retrieval matches that at learning. For example, Godden and Baddeley (1975) got divers to learn material either on dry land or underwater and to recall them either underwater or on dry land. Recall was best when the context at recall matched that at encoding. In another famous study Carraher, Carraher and Schliemann (1985) showed that Brazilian street children were able to perform complex mathematical calculation in the street but not in a school context.

Another form of unguided learning includes experiential learning theory (ELT), which claims to provide “a holistic model of the learning process […] consistent with what we know about how people learn, grow and develop” (Kolb, Boyatzis, & Mainemelis, 2001, p. 193). Observing and reflecting on the effects of concrete experiences and actions lead one to assimilating these reflections as abstract concepts. These in turn lead to implications that can be tested to see what would follow and to serve as guides if the same actions were to be repeated (see Figure 4.7). The learner has choices about how to go about engaging with each stage since individual differences in personality and experience lead to differing learning styles.

Constructing or discovering knowledge has been the basis for problem-based learning (PBL). This follows a generally constructivist approach where students are required to solve an open-ended question, such as diagnosing and finding a treatment for an ailment through their own investigations. This example reflects the fact that PBL was originally promoted in medical schools in North America. The aim is to develop problem solving skills, lifelong learning skills and those involved in working in a team.

There was a debate in the late 1990s about aspects of instructional design and the extent to which knowledge and skill can transfer from one domain or context to another. Some researchers have argued that production models of learning involving condition–action rules such as ACT-R (Anderson, 1983) tend to ignore metacognitive skills and the social contexts of learning (although see Taatgen, Huss, Dickison, & Anderson, 2008). Brown and Palincsar (1989), for example, stated that students learn specific knowledge-acquisition strategies. Group settings encourage understanding and conceptual change and these are contrasted with “situations that encourage automatization, ritualization, or routinization of a skill, in which speed is emphasized at the expense of thought” (Brown & Palincsar, 1989, p. 395). Cobb and Bowers (1999) also argue that the cognitive approach to learning adopted by Anderson (among others) may not be helpful in deciding what to do in a classroom. Figure 4.8 gives an indication of the emphases placed on different aspects of the learning by those espousing a “cognitive” position, such as Anderson, and those espousing a situated learning position, such as Greeno (e.g., Greeno, 1991). In the top half of the figure, knowledge is seen as something we have which, given a particular task environment and the limits of our information processing system, leads us to our behaviour. In the bottom half, knowledge is seen as an activity generated by our social, cultural and physical context and manifesting itself as overt behaviour.

Anderson, Reder and Simon (1996, 1997) reject a strong version of the claim that all knowledge, both specific and general, is “situationally grounded” and hence does not readily transfer. Learning arithmetic in school does not mean that you can’t make calculations in a supermarket. Similarly reading and writing can take place in a wide variety of contexts. Reading, for example, is not normally taught while lying on a beach but we can manage to read books in that context nevertheless. There are, however, aspects that can be agreed upon, hence Anderson et al. joined with James Greeno to produce an account of the complementarity of situational and cognitive approaches to learning (Anderson, Greeno, Reder, & Simon, 2000). However their critiques of aspects of situated learning continued in the same year (Anderson, Reder, & Simon, 2000). With regard to situated learning they argued that the degree of contextualisation depends on how the material is taught and whether it is taught in narrow, specific contexts. If material is context bound then transfer is unlikely, however Anderson et al. argue that there are many studies showing a lot of transfer, little transfer or negative transfer depending on the manipulation of the study. Situated learning theorists expect apprenticeships to offer the best form of training rather than the more abstract learning fostered by schools. However, presenting abstract concepts alongside concrete examples and illustrations can be very effective (see, e.g., the arguments by Ross discussed earlier). Finally, Anderson et al. criticise the view that learning must take place in a social context – the “communities of learning” view. It is sometimes useful to learn to work with others and sometimes to work by yourself. Writing books, for example, can be a lonely process.

Anderson et al. also criticise aspects of theories based on constructivist principles including the following:

• “Knowledge cannot be instructed (transmitted) by a teacher, it can only be constructed by the learner” (p. 39). However, the role of the teacher is to help students engage in activities they would not otherwise engage in. They also help students save a lot of unnecessary time in lengthy search which can be demotivating.
• “Knowledge can only be communicated in complex learning situations” (p. 44). Anderson et al. argue that someone learning to play the violin would have a hard time doing so if all their practice was in the context of an orchestra. The processing demands would be too great.
• “It is not possible to apply standard evaluations to assess learning” (p. 45). Anderson et al. quote Confrey (1991) when referring to students as potentially being the best judges of their learning:

And some of these students can also be completely wrong.

While the various flavours of constructivist and discovery-based learning rely on learners finding things out for themselves with greater or lesser guidance, Mayer (2004) and Kirschner, Sweller and Clark (2006) have argued that students’ learning is much more efficient and effective if you explain things to them. They, along with Anderson et al., argue that there is little empirical evidence that unguided and minimally guided approaches are effective, and both groups argue for empirical studies comparing the various educational programmes.

Mayer (2004) does not object to the view that learners construct their own knowledge or that learning should be “active”. He does object to the view that being “cognitively active” is the same as being “behaviorally active”, and refers to this as the “constructivist teaching fallacy”. He reviewed various topics based on versions of discovery learning in the ’60s, ’70s and ’80s, where discovery learning manifested itself in various ways and where empirical evidence showed that pure discovery learning was ineffective compared with guided discovery learning. He argues that since discovery methods of instruction have been repeatedly shown to be ineffective, there is no point in continuing to return to this methodology.

In similar vein, Kirschner et al. (2006) have argued that unguided or minimally guided approaches to instruction tend to ignore human cognitive architecture and its consequences for instructional design.

Kirschner et al. do not take issue with the view that learners construct knowledge by generating mental representations or schemas in long-term memory, but, like Mayer (2004), they do take issue with the instructional approach used, which takes the view that instruction means experiencing the procedures involved in whatever discipline is being studied. They claim that teachers attempting to provide instruction based on constructivist principles often end up providing guidance, especially when students fail to progress. Again like Mayer, they claim that there is little evidence that unguided and minimally guided teaching methods are effective and that the worked example effect (Cooper & Sweller, 1987; Sweller & Cooper, 1985) demonstrates the superiority of guided instruction over forms of unguided instruction.

However, Hmelo-Silver, Duncan and Chinn (2006) have criticised Kirschner et al. (2006) for lumping in problem-based learning and inquiry learning (IL) with other forms of minimally guided learning and discovery learning. They argue that PBL and IL are not discovery approaches or examples of minimally guided instruction due to the wide array of scaffolding provided (patient data, research papers, scientific materials, web-based information sets, software, teachers, etc.). They also argue that there is evidence of improvements in learning as well as motivation and engagement compared with other pedagogical methods. Unfortunately for their argument Clark, Kirschner and Sweller (2012) continued to include PBL and IL as examples of partially guided instruction 6 years later.

Marra, Jonassen, Palmer and Luft (2014) also make the case that PBL works. They argue that it is based on constructivist principles and that it comes under the rubric of situated learning as the students face problems in real-world contexts. They emphasise the role of cases (e.g., medical cases) used as instructional examples. Instructors or facilitators can point to already solved cases to use as potential analogues. Thus much of PBL can be seen as a form of analogical problem solving (case-based reasoning: see also Hernandez-Serrano & Jonassen, 2003; Jonassen & Hernandez-Serrano, 2002).

Another comparison between instructional approaches has been made by Chen et al. (2015). They explain the apparently contradictory results shown by studies of the worked example effect involving a high level of guidance (where worked examples produce better performance than getting students to simply solve problems), and the generation effect where students who generate their own responses (with a low level of guidance) perform better than those who simply study answers to questions. Chen et al. explain the difference by reference to the degree of element interactivity. The worked example effect predominates when there is high element interactivity whereas the generation effect works best when there is relatively low element interactivity. When the load on working memory is already low, any attempt to reduce the load turns out to be counterproductive, hence the performance boost for the generation effect.

Providing a schema in texts to aid understanding

As has been discussed, one method of providing the reader with help in understanding new concepts is by providing an explanation alongside a worked example, thereby providing an explicit explanatory schema rather than leaving the schema implicit. Examples can represent a category of problems and it is therefore possible to provide a general schema for solving a range of problems of the same type. As we saw in Information Box 4.4 (Reed et al., 1985), the difficulty here is presenting the schema at the appropriate level of abstraction. If it is too abstract, then it might be hard to see how it applies in a specific example. If it is too specific, then it might be hard to see how it can be transferred to another more distant variant of the problem type. In general, however, it appears to be the case that providing explicit conceptual instruction prior to problem solving is more effective than other forms of instructional sequence (see e.g., Fyfe et al., 2014; Hsu, Kalyuga, & Sweller, 2015). Smith and Goodman (1984) have summarised the benefits of providing an explanatory schema (see Table 4.4).

Table 4.4 The benefits of an explanatory schema and of diagrams

Schemas Smith and Goodman (1984)	Diagrams Larkin and Simon (1987)
Schemas provide an explanatory framework or “scaffolding”. They improve understanding since the pre-existing connections between the framework slots can be mapped to the new domain directly.	In Larkin and Simon’s terms the diagram and text should be “informationally equivalent” so that information in one representation is also inferable in the other.
Schemas contain information that can be added to fill in gaps in knowledge and help form connections between steps.	In diagrams this includes the ability to generate perceptual inferences.
Schema-based instructions reduce the time required to understand the relation between steps.	In diagrams there is less need for search.
Schemas boost memory for specific information.	According to Larkin and Simon, in diagrams perception permits the reader to focus on perceptual cues and so retrieve problem relevant inference operators from memory.
Schemas boost performance where they depend on understanding the relations between steps.	Similarly, diagrams have computational benefits, since the information in them is better indexed and is supported by perceptual inferences (Moreno et al., 2011).
Schemas should lead to a hierarchical organisation of material which should, in turn, lead to “chunking” and hence to improved recall (Eylon & Reif, 1984).	The information in diagrams is perceptually grouped – related bits of information are adjacent to each other.

Providing a picture in texts to aid understanding

An alternative to providing an explanation of a concept or abstract principle is to provide a visual representation in the form of an image or diagram such as the ones throughout this book. They can be concrete (an image of seas, mountains, clouds, rain, sunshine and a few arrows can represent the hydrological cycle), abstract (such as graphs or electrical circuits) or both. Moreno, Ozogul and Reisslein (2011) used a combination of abstract circuits and ones with images of a battery and light bulbs. One condition had diagrams of electrical circuits, one had diagrams but with images of a battery and light bulbs, and one had both on the same image. Students who received instruction using both outperformed those receiving abstract only and those with concrete images on a number of measures.

The reasons why pictorial or diagrammatic representations can be effective are discussed by Larkin and Simon (1987). Texts present information in a linear sequence. Understanding this sentential representation incurs a great deal of computational cost in terms of search. The larger the data structure contained within the sentential representation, the greater the search time. It is as if you had to search through a lot of “mental text” to retrieve relevant information or make a useful inference. In a diagrammatic representation the information is indexed by its two-dimensional location, thus diagrams can make relations perceptually explicit, which is not the case in sentential representations. According to Larkin and Simon, diagrams:

1. 1  Allow a large number of automatic perceptual inferences;
2. 2  Avoid the need to match symbolic labels (matching a variable in one part of a sentential representation to a related variable elsewhere);
3. 3  Obviate the need to search for problem solving inferences.

The relative merits of schemas and graphical representations are covered in Table 4.4.

While graphical representations can help us understand concepts or systems, they also have a number of other functions. Levin (1988) classifies the functions of “pictures-in-text” into five categories:

1. 1  Decoration, where pictures are designed to make a text more attractive but are not related to the content;
2. 2  Representation, where pictures make the text more concrete, as in children’s books;
3. 3  Organisation, where pictures enhance the structure of a text;
4. 4  Interpretation, where pictures are supposed to make a text more comprehensible;
5. 5  Transformation, where pictures are presented to make a text more memorable.

Levin relates these functions to different prose-learning outcomes by appealing to the notion of transfer-appropriate processing (Morris, Bransford, & Franks, 1977). Learners have to take account of the goals of the learning context and adapt their learning strategies accordingly. In the context of using pictures in text, Levin argues that writers should use different pictorial representations depending on whether they want to encourage the learner to understand the material, remember the material, or to apply the material. For example, in the studies by Beveridge and Parkins (1987) and Gick (1985) the function of the graphical representation was principally to aid retrieval.

With regard to using graphical representations to understand the material, Cheng (2002) presented students with either traditional algebraic representations of electrical circuits or a novel type of electrical diagrams (AVOW: Amps, Volts, Ohms, Watts). He found that problem solving using the diagrams was more effective than using algebraic manipulations at helping students solve complex transfer problems. Essentially the diagrams reduced the cognitive load imposed by high element interactivity; or in Cheng’s words, the students “acquired coherent networks of concepts” (p. 721). Cheng hypothesised that “a representational system that makes the nature of the domain directly apparent in the inherent structure of the representation itself will be likely to enhance learning by making interpretations of laws and cases easier” (p. 722).

Conclusion

Given what we know about human cognition, it ought to be possible to support students’ learning in all sorts of ways. There are, however, many disagreements about the best approaches to instruction. Nevertheless, we know enough to be able to state some instructional design principles with a degree of confidence. Mayer and colleagues (Mayer, 2001; Mayer & Moreno, 2003; Moreno & Mayer, 2007), for example, have listed a number of principles of instruction using multimedia (pictures, text, narrations, animations) that get round some of the limitations of working memory and cognitive load. Words and pictures both pass through the eyes and so would usually interfere with each other; narrations along with pictures usually won’t. Words can conjure up images and so can contribute to a pictorial model of the to-be-learned material as well as a verbal model. By sticking to these principles, a writer or teacher should be able to reduce extraneous cognitive load, for example by making sure that pictures and words are presented together so that verbal and pictorial models are available in working memory. The first part of Information Box 4.5 describes Mayer’s principles of multimedia instruction and the second part includes further strategies for enhancing learning.

Summary

1. 1  Analogical problem solving involves reasoning from a familiar domain to solve problems in an unfamiliar one. Textbook problem solving, on the other hand, tends to involve mapping an unfamiliar example onto an even less familiar exercise problem.
2. 2  Textbook writers have to make some assumptions about the readership. These include assumptions about:
   • The readers’ prior knowledge;
   • How much the readers are likely to remember from previous chapters;
   • How much they understand from previous chapters;
   • Their schema knowledge of how such texts are constructed;
   • How much the readers can generalise from the examples and explanations given.
3. 3  Examples in textbooks are the salient aspects of instruction. They are the parts of the text that readers pay most attention to and use when solving later problems. They show:
   • How abstract principles can be made concrete;
   • What operators to choose at any given point;
   • What features can readily be generalised over.
4. 4  Different forms of representation are necessary to understand word problems. These include knowledge of the lexicon, understanding of the text including local coherence (the textbase), the mental model of the situation described in the text including inferences derived from it (the situation model), and the relation between the latter and the solver’s knowledge of the domain (e.g., mathematical knowledge) that allows the generation of the problem model.
5. 5  When solvers attempt to use an example as a source to solve a target problem they are likely to be successful if the two are close variants. If they are distant variants then the textual explanation needs to include an explanation of how to generalise over the different variants (i.e., a problem schema needs to be provided).
6. 6  Cognitive load theory provides a theoretical background to the design of instruction based on the fact that working memory has a limited capacity. Dealing with problems involves mental effort, and teachers need to reduce the load on the cognitive systems of students by being aware of the different forms of cognitive load. These are:
   • Intrinsic: the inherent difficulty of the to-be-learned material;
   • Germane: the load needed to process the to-be-learned material;
   • Extraneous: the load imposed by the way the material is presented.
7. 7  There are differing views about the effectiveness of guided learning versus unguided or minimally guided learning. There is agreement about the need for the student to be actively engaged in knowledge construction but disagreement about the pedagogical methods needed to achieve it.
8. 8  Diagrams and analogies provide a means of forming a bridge between a familiar situation and the novel unfamiliar situation. Although pictures, graphs and illustrations can have a variety of functions in texts, they can share the same pedagogical function as analogies and schemas in texts.