Cooking shows. How-to videos. Dance choreography. Test-prep books. Math classes. All of these things share one key attribute: worked examples.
What it is.
You remember these, don’t you? Every math textbook that I have ever seen has these things: worked examples.
Most of the worked examples in the research literature follow this kind of format: a problem — mathematical, scientific, or technical — broken down into steps to solve it.
But you might consider a broader definition of worked examples to include many other things. Demonstrations often function as worked examples: let me show you how to change a tire on a bike. Or let me show you how to follow this choreography.
Worked examples are everywhere. It’s not one of those “secret learning hacks” that you hear about on Youtube. The bigger question is: how do worked examples facilitate learning? And what are the biggest mistakes people make when using them?
How it works.
There are several explanations for the effectiveness of worked examples.
The earliest explanation was that worked examples are fundamentally about cognitive load. Cognitive load is about what people pay attention to and mentally process. When a student solves a problem, they might be paying attention to any number of different things that are irrelevant to what they’re trying to learn. They can fall prey to misconceptions and wander down pointless paths. When a student examines a (good) worked example, however, the student focuses on what’s most relevant for them to learn. So students pay attention efficiently.
A slightly different twist on this same explanation focuses more on how worked examples illustrate expert thinking — especially when a student doesn’t know at all what to do. The more that worked examples illuminate expert thinking, the more effective the examples tend to be.
A final explanation is that worked examples alter the orientation of the learner. It’s the difference between playing a game of Starcraft and analyzing a Starcraft game after-the-fact. During a competition, the priority is winning. During game analysis, the priority is learning. The same thing goes on when students are actively solving problems — students focus on what to do next to solve the problem. They’re interested in getting the answer. But as teachers, we don’t really care whether students got the right answer; we care whether students have learned the right concept. Worked examples emphasize the process: what are the steps and how do we solve this kind of problem?
How we know it works.
The most common way worked examples have been studied is through controlled laboratory research.
The earliest research focused on comparing worked examples to unguided problem-solving. The goal was for students to learn an algorithm for solving the problem. Students in the worked example condition would learn how to solve a problem by — naturally — looking at worked examples. Students in the control condition would try to solve those same problems on their own, without the same information, and without any guidance.
Now this comparison was a little silly, as very few people seriously argue in favor of this kind of “pure” discovery learning (i.e., “I’m not going to tell you anything about this… just go learn it!”). Nevertheless, worked examples worked. And they have continued to demonstrate their value in more rigorous control conditions.
The more interesting research on worked examples is on the limits and effective use of them. Worked examples are most effective when someone knows nothing about the domain. You’ve never solved an algebra problem before: here’s a worked example.
As students learn more, worked examples can be ineffective, and even counter-productive. The exact reason for this “expertise-reversal effect” is not clear, but the “extra” information may be confusing more experienced learners because they look at the worked example expecting it to contain information that they don’t already know — but it doesn’t.
Worked examples alone also can’t make someone a good general problem solver. The focus is on applying a procedural “schema” or pattern onto a certain class of problems. Other types of examples — for instance, erroneous examples or contrasting examples — can work to deepen understanding.
How to implement it.
Bad worked examples abound. In college, I had a math teacher who spent all class with worked examples. He would take a complex problem, and walk through the steps on the blackboard. But we learned almost nothing from this exercise. Although we could see the steps he was making, he didn’t explain them. When students asked questions, he mumbled through answers, leaving us more perplexed than before.
Effective worked examples illuminate expert thinking; they don’t obscure it. This is why worked examples that include reasons for the steps (or cues to encourage students to think about those reasons) help students more than worked examples that do not. Consider identifying sub-goals for the students first, rather than just rattling off the steps you took.
Effective worked examples also go away. The goal of worked examples is to help novice students gain procedural fluency. Once they get a handle on some of the techniques and common situations, more worked examples won’t help them. All of the extra information that was helpful to the novice is distracting. It becomes “regular practice”. At this point, it’s better to transition students into more realistic problem-solving scenarios.
There are also some risks in asking students to follow fixed procedures. For example, it’s often helpful to draw a diagram when solving physics problems. This helps problem solvers conceptualize what’s going on, interpreting text into a model of the situation. Makes sense to tell students to draw these diagrams — it’s an expert practice and a good first step at solving the problem. Doing so, however, can have some perverse effects. When researchers ask students to draw a physics diagram, they are actually less likely to solve problems in an insightful way.
What seems to be happening is that asking students to draw a diagram puts them into a procedural frame of mind. This frame of mind blocks more creative, insightful approaches. If students are solving math and science problems in this way, it can also make it seem like math and science is mostly about procedural drudgery (and not about thinking conceptually).
One potential way of avoiding this problem is to provide contrasting worked examples which illustrate how similar-looking situations might be solved differently. Or to illustrate how different-looking situations might be solved in the same way. Nothing stops a worked example from showing how an expert problem-solver would categorize the problem, instead of solve it. Or the examples might explain the route of a typical mistake. These are all more challenging, advanced uses of worked examples, however, and best used after students have some familiarity with the problems.
There are other ways to make worked examples even more effective. Fading out the steps to a worked example is an effective technique. Interleaving worked examples with open problem solving can also improve learning outcomes. The problem solving motivates and contextualizes the information contained in the worked examples, so that students interpret that information more effectively.
Like many other instructional techniques, worked examples is a tool: powerful when used wisely; counter-productive when used unwisely.