Contents.Brief statement of principleIn contrast to the traditional approach of giving a list homework (or seatwork) problems for students to solve, students learn more efficiently and more robustly when more frequent study of worked examples is interleaved with problem solving practice.Description of principle'In courses that are teaching new tasks, learning time can be saved by replacing some practice problems with worked examples' (Clark & Mayer, 2004, p. In addition, most studies comparing interleaved worked examples and problems with all problems have also shown improved learning outcomes, including robust learning outcomes.' It would be an unusual (not to mention incompetent) teacher who did not use worked examples.
Similarly, textbooks universally use worked examples to illustrate new concepts. The suggestion being made here goes beyond this limited use of worked examples. Rather than using them merely to demonstrate how to use a mathematical or scientific rule, the proposal is that they should be used in large numbers as a form of practice. In other words, instead of practicing by solving many problems (an activity engaged in by most conscientious students), it is proposed that many of these problems could profitably be replaced by worked examples.' (1999) p73Operational definition ExamplesImagine instead of giving students a typical homework or seatwork assignment involving 8 problems, you give them an assignment where every other problem comes with a complete worked out solution. The even numbered items would be usual problems, like the following algebra problem:Solve 12 + 2x = 15 for xThe odd numbered problems, come with solutions, like this:Solve 12 + 2x = 15 for xStudy each step in this solution, so that you can better solve the next problem on your own:12+2x = 152x = 15-122x = 3x = 3/2x = 1.5Which approach, asking for solutions to all 8 problems or interleaving 4 examples with 4 problems, will lead to better student learning? You might think that the 8 problems require more work or that students might ignore the examples and thus, the 8 problems would lead to more learning.
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But, much research has shown that students typically learn more deeply and more easily from the second approach, when examples are interleaved between problems.Teachers often think so many examples “give it away” or that students will not pay attention to the example. But, by having problems in between students are motivated to pay more attention to the example so as to prepare for the next problem or to resolve a question from the past problem. The problems break a students’ “illusion of knowing” that might otherwise lead them to skim the example and believe it is obvious.It is important that students spend time actively engaged in learning and in genuine problem solving and reasoning. However, an emphasis on “learn by doing” is sometimes taken too far and students end up with homework problems or projects that are beyond their means.
In such cases, they may spend much unproductive study time struggling without success. This time is often not only wasted but may increase a students’ frustration with the subject-matter and lead to unjustified feelings of not being good at math or science particularly.
In contrast, during example study, students can focus their attention on understanding the principles underlying the examples instead of simply on finishing the problem. In early learning, the thought that goes simply into trying to solve the problem seems to distract students from trying to understand the principles underlying the solution.Notice that in the example above, explanations for each step are not provided. It is best when students provide these explanations themselves (see the ) and, while more research is needed, providing explanations can sometimes distract students from doing so themselves and in other cases seems to provide no additional enhancement in student learning.In whole classroom situation a teacher might implement this principle by going back and forth between a classroom or small group discussion around an example solution followed by small groups or individuals solving a problem (just one!) on their own. Then back to example study, for instance, by having students present their solutions and having others attempt to explain the steps (see the ). Now back to a second problem.By giving the students frequent opportunities to study examples in between problem solving, students can more easily and more deeply acquire the big ideas, key concepts, or key principles that we want them to learn.
With greater understanding, students will do better on harder problems in the future that require them to transfer these key concepts beyond the problems just like those they have seen before.Experimental supportAs summarized in Clark & Mayer, 2003 (pp 179): 'There is a lot of evidence for the effectiveness of learning from worked examples. As an example, in one study twelve statistics problems were used. In the conventional group the learners solved all twelve problems as practice. In the worked examples group, the learners received eight problems already worked out to study and then four problems to solve as practice. Students in the worked examples group spent significantly less time studying and scored higher on a test than did those in the conventional group.
Furthermore, the worked examples group scored higher not only on test problems similar to those used during practice but also on different types of problems requiring application of the principles taught (Paas, 1992). The investigators conclude that 'training with partly or completely worked-out problems leads to less effort-demanding and better transfer performance and is more time efficient' (p.
In fact, in one study, the use of worked examples allowed learners to complete a three-year mathematics course in two years (Zhu and Simon, 1987). Positive effects of worked examples have been reported in a variety of courses teaching well-defined problems, including algebra, geometry, statistics, and programming'.Laboratory experiment supportSee papers Cooper & Sweller and many others, such as Atkinson, Ayres, Catrambone, Paas, Renkl, van Gog, van Merrienboer. See the recommendation 'Repeatedly alternating problems with their solutions provided and problems that students must solve' which is a web-based update of this IES practice guide: Organizing Instruction and Study to Improve Student Learning. Some relevant references can be found withinIn vivo experiment supportprovide mixed support of the worked example principle. Although students did not learn more through the study of worked examples followed by problem solving, as in (Paas, 1992; Zhu and Simon, 1987; Trafton & Reiser, 1993), they did learn more efficiently as in the earlier studies. On the other hand, only normal pre-post gains were evaluated in the stoichiometry studies; was not measured.
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In addition, the control condition of these three studies was different - and potentially much more rigorous - than the earlier studies: students solved problems with the support of an intelligent tutor. This may explain why students did not learn more: perhaps the additional support of the tutor - in which students theoretically could create their own 'worked examples' by clicking through to bottom out hints - equalizes the advantage of learning from the examples.(2008) studied the use of 'faded' examples as an adjunct to tutored problem solving with the Geometry Cognitive Tutor. Following Renkl and Atkinson's (2003) example fading methods, the example-enhanced version of the tutor, after first presenting fully-worked-out examples, gradually reduced the number of solution steps given, thus increasing the number of open steps that students had to solve. Salden et al.