Worked examples

Problem solving is described as a process where a learner is attempting to find ways to change from the original state of knowledge about a problem to the required solution, or end state of knowledge, for the problem (Newell & Simon, 1972). This process is also known as a means-end search as the process of problem solving involves identifying the goal and the end state they wish to achieve, using the means to reach the ends. However, for more complex problems, the desired end cannot be obtained from a single process; as such, the subjects have to establish sub-goals that help in achieving their desired goal.

Sweller (1988) suggested that there may be a degree of incompatibility between the process of learning and problem solving. A learner would need to engage the means- end analysis before arriving at the solution of the problem. CLT argues that the commonly used search processes (such as the means-end analysis) frequently use different cognitive processes for learning each of the sub-goals. Therefore, the search strategies may direct attention to different aspects of a problem and have the potential to use all available WM and reduce the effectiveness of the learning process. Sweller’s (1988) computational model indicated that problem solving through means-end analysis has a heavy cognitive load. For a learner, if search strategies require different cognitive process to those required for learning then these strategies have the potential to increase cognitive load and, thus, WM resources are less likely to be available for the process of learning.

Worked examples have been put forward as one technique designed to reduce ECL interference with learning that is caused by some forms of problem-solving techniques. Worked examples describe the problem statement and all the necessary steps to solve the problem. Unlike conventional problem-solving techniques through means-end analysis, worked examples direct the attention of a learner to the problem and state the steps required to solve a particular type of problem. Learners can benefit more from studying worked examples than by solving equivalent problems through means-end analysis (Trafton & Reiser, 1993). As such, the ECL from worked examples has the potential to be less than that from the conventional problem-solving methods. These reductions in cognitive load can possibly make the process of learning easier for a learner. According to Jelsma et al. (1990), learning with the aid of worked examples can be more effective in problem solving than learning from solving the actual problem itself. Sweller and Cooper (1985) and Cooper and Sweller (1987) studied the use of worked examples for learning algebra compared to using the conventional problem-solving method. Sweller and Cooper (1985) argued that the use of worked examples improved the learner’s ability to construct a method for solving algebra problems and also improved their ability to transfer their knowledge to solving related algebra problems. Thus, through the use of worked examples, greater learning outcomes were achieved. Figure 2.3 below expresses an algebraic equation demonstrated through a worked example shown in Sweller and Cooper (1985 p.70).

For the equation a = ag +b, express a in terms of the other variables.

𝑎 = 𝑎𝑔 + 𝑏 𝑎 − 𝑎𝑔 = 𝑏 𝑎(1 − 𝑔) = 𝑏

𝑎 = _{1 − 𝑔}𝑏

However, Sweller and Cooper’s (1985) investigation into the use of worked examples as a instructional technique identified one possible limitation, that they may not provide sufficient motivation for learners. Learners have the potential to just read worked examples and not use them as an effective way to develop the knowledge within the example. To overcome this limitation Sweller and Cooper (1985) altered the worked examples, using a structure identical to the problem exercises given to learners. It was expected, for a learner to develop adequate schemas, that the schemas were able to be constructed more readily from studying the examples and then solving problems, rather than a learner just being provided with simple problems to study. Sweller and Cooper (1985) identified that those who learned from worked examples were able to attempt the problem faster during a test than those who only learned through conventional problem-solving techniques. Further experiments into worked examples were conducted to evaluate if they could be used for the transfer of knowledge, with similar and dissimilar problems allowing schema automation to occur. To test the use of worked examples to indicate knowledge transfer to similar or dissimilar problems, Sweller and Cooper (1985) designed several studies. The results of these experiments indicated that knowledge from studying worked examples was useful when solving similar problems but not for dissimilar problems. However, more intensive use of worked examples was beneficial in the transfer of knowledge to dissimilar problems. Kotovsky et al. (1985) also emphasised that extra practice time is necessary for schema automation, which is the key factor in knowledge transfer. Zhu and Simon (1987) also confirmed, from their experiments, that learners acquired schemas and achieved automation more efficiently through the use of worked examples than through conventional teaching methods. Their findings revealed that a three-year mathematics course could be completed in two years by students who were taught using worked examples; their analysis of protocols noted that students were more engaged in learning using worked examples. Therefore, when using worked examples in a learning environment, it is important to consider the amount of time that a learner is given to develop their knowledge for this method to be successful in improving learning outcomes.

Paas (1992) and Paas and Van Merriënboer (1994) conducted experiments where they compared groups of students learning through using worked examples compared to those who learnt through a combined method of both conventional problem solving and worked examples. Paas (1992) used problems from statistics whereas Paas and Van Merriënboer (1994) used problems from geometry. The results indicated that the groups studying from worked examples alone had higher construction of schema and transfer performance and lower ECL in comparison to the group that used the combined method. The reason for lower effectiveness of the group using combined method was the degree of failures in attempting to find the solution from their conventional problem solving into the schema they acquired for solving the problems.

Another study by Novick (1988) indicated that worked examples facilitate learning and problem solving more than solving equivalent problems. His research also highlighted that experts use worked examples differently to novices, in order to solve problems, and focused more on structural features of worked examples than novices, which is more relevant in solving transfer problems.

Further research by Chi et al. (1989) focused on effective and ineffective learners as they studied with worked examples. Their findings also indicated that students taught using worked examples were more able to produce in-depth details and had greater awareness of the failures. Therefore, this research indicates that good learners can explain worked examples and identify the main problem features and operators, which may also assist ‘self-explanations’ (Chi et al., 1989; Renkl & Atkinson, 2002). Sweller et al. (1998) summarised the research on worked examples and concluded that worked examples are beneficial for learning outcomes and transfer. In summary, learners in practice often view worked examples rather than explanatory notes as a more relevant source for learning materials (Lieberman, 1986; Pirolli, 1991; Segal & Ahmad, 1993). Thus, greater learning outcomes are achieved.