The Power of Worked Examples
Brodie: Hey everyone, I’m not sure if this is allowed, but I’m completely stuck on questions 3,4, and 5 of the assignment for 'Mechanics of Materials 201'. I’ve already read them several times, but I haven’t made any progress. I learn so much better if I can see the entire process. I’m not looking for the answer but can you provide me with a detailed step by step?
When it comes to STEM type subjects like Engineering, Maths, Physics, Chemistry etc, solving problem sets is one of the main forms of assessment whether in assignments or exams. Even if the questions in the exam are multichoice the answers often consist solely of numerical options meaning some sort of calculation is required. Hence it is critical to become proficient at solving problems.
In Brodie’s case a likely scenario is that he is trying to tackle the assignment without having studied any similar examples beforehand or attempted any similar problems for himself. For many students the first time they attempt to solve a problem of a particular type is when they see it in an assignment. This isn’t a good approach to learning. Starting to learn the material early gives more time to overcome difficulties. It’s also easier to get help from instructors when they aren’t flooded with questions right before an assignment is due.
Unless you are at an elite institution with a tough instructor, it is extremely rare that an assignment problem will be something especially novel or complex. Instead, assignment problems are typically similar to what is found in the textbook, course notes, and other assigned instructional materials.
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Tip
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Try solving problems of a particular type prior to encountering them in an assignment. |
When studying a new topic the traditional approach is that the instructor explains the theory, shows a few worked examples, then expects the student to dive into solving large numbers of problems for themselves. The issue with this approach is that some students are unprepared for university, and are unable to make an transition to solving open-ended problems. If Brodie is struggling to solve problems for himself, how can he overcome this? One approach is effective use of worked examples.
Trying to keep in mind all the information needed to solve the problem can overwhelm a student’s cognitive resources. Studying a worked example can let them focus on learning the main procedural steps, while not getting too distracted by the details. The evidence points to worked examples being most useful for novice learners [1]. As students' become more proficient, and understand the principles, they can tackle problems independently.
Some studies suggest that early in the learning process, given a fixed amount of time for study, reviewing more worked examples enables additional learning compared to problem-solving practice alone [2]. But as expertise improves, studying examples becomes less important, and time should be allocated increasingly to problem-solving [2].
Another scenario where worked examples can be useful is if solving problems is impractical. This could occur if you are commuting using public transport for example. In this situation reviewing a few worked examples can be a good supplemental study activity.
Generally speaking the more problems a student can be exposed to the better. When students have solved a certain number of problems for themselves, and are starting to get fatigued and run out of time, adding the study of a few worked examples to the end of the study session can be a way to boost the total number of problems a student is exposed to.
Although not ideal, it is possible to pass a course even if you don’t understand the theory that well, provided you have a sufficient understanding of the steps needed to solve the relevant problems.
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Tip
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Start by studying worked examples, then a mixture of open-ended problems and examples, then finally mostly problems. |
Faded examples
Sometimes students get overwhelmed when attempting to solve a novel problem and have trouble getting started. One effective technique to deal with this is the use of faded examples. In the first faded example the problem is completely worked out. The next one has every step completed except for the last one which the student needs to work out. Each subsequent faded example has one fewer step completed until eventually the student is able to work out the complete problem for themselves.
The idea is to provide scaffolding, and then progressively removing this scaffolding as the students' skill improves. Students report finding this approach useful, and there is some evidence that it helps unprepared students catch up to their better prepared peers [3].
By having less steps for the learner to complete early on, their cognitive capacity doesn’t get overwhelmed, and they are able to focus on the overall concepts. A number of studies have shown benefits for faded worked examples such as becoming proficient in less time than using exclusively using open-ended problems [4].
Where to find worked examples?
If the instructor doesn’t provide enough worked examples, consult the prescribed textbook, or other textbooks that cover the same subject. In addition to the textbooks themselves, there are often associated solution manuals or workbooks that contain a lot of complete problems. There are also supplementary books that contain large numbers of worked examples such as the well known Schaum’s Outline series. Of course there are also numerous websites that feature notes and work examples.
In the extreme case of really not having access to enough examples, an AI tool can often be used to produce a worked example based on a question you’ve supplied. Normally AI does a decent job, but obviously you should exercise caution as there is a risk it will get it wrong. This can be mitigated to some extent because most textbooks give the final answer, but not the steps, for some of the exercises.
A bad way to use examples
When I was a student one calculus lecturer was so bored with teaching that he never prepared ahead of time for the lecture. Instead he would turn up with the textbook and copy the main theorems for the topic onto the whiteboard. Next he would pick the most difficult problem from that section of the textbook, and attempt to solve it. The example was so long and complex that it resulted in two issues. Firstly, it obscured the basic principles and overwhelmed most students in the class. Secondly, he inevitably made a mistake somewhere, and the answer didn’t match what was in the back of the textbook. The textbook didn’t have worked solutions for the problem sets, just the final answer. By now the alloted time for the lecture was over and he was frantically trying to identify the location of the error. The bulk of the lecture was wasted on a worked example that nobody really followed or was able to learn much from. Studying it later could be confusing because of one or more mistakes in unknown locations.
When first encountering a topic, don’t use examples that are too complex. A simple one that shows the main principles is a better starting point. As you gain proficiency you can build up to more complex problems.
An incorrect example is not always bad, provided it is carefully designed. An example that illustrates a common error students' make is valuable. One or more random errors in unknown locations of an example is unhelpful and confusing.
Summary
Learners only have limited cognitive resources. In the early stages of learning a topic, these resources can be overwhelmed. Studying a worked example can assist with focusing on the main ideas. If you are struggling with a certain topic, seek out additional worked examples. Use faded examples to help get you started. Once you can solve problems independently, boost the total number of problems you are exposed to by mixing in additional worked examples.
References
[1] J. L. Booth et al., “Evidence for cognitive science principles that impact learning in mathematics,” in Acquisition of complex arithmetic skills and higher-order mathematics concepts, Elsevier, 2017, pp. 297–325.
[2] C. A. Barbieri, D. Miller-Cotto, S. N. Clerjuste, and K. Chawla, “A meta-analysis of the worked examples effect on mathematics performance,” Educational Psychology Review, vol. 35, no. 1, p. 11, 2023.