Teach by Example(s)?

Consider this story:

A general wishes to capture a fortress located in the center of a country. There are many roads radiating outward from the fortress. All have been mined so that while small groups of men can pass over the roads safely, any large force will detonate the mines. A full-scale direct attack is therefore impossible. The general’s solution is to divide his army into small groups, send each group to the head of a different road, and have the groups converge simultaneously on the fortress.

Cool story. The general’s solution is practical and logical. Now check out this scenario given to students:

Suppose you are a doctor faced with a patient who has a malignant tumor in his stomach. It is impossible to operate on the patient, but unless the tumor is destroyed the patient will die. There is a kind of ray that can be used to destroy the tumor. If the rays reach the tumor all at once at a sufficiently high intensity, the tumor will be destroyed. Unfortunately, at this intensity the healthy tissue that the rays pass through on the way to the tumor will also be destroyed. At lower intensities the rays are harmless to healthy tissue, but they will not affect the tumor either. What type of procedure might be used to destroy the tumor with the rays, and at the same time avoid destroying the healthy tissue?

Both of these scenarios became famous when Gick & Holyoak published a study called Analogical Problem Solving that investigated ways learners can benefit by examining problems that are similar to each other. When students were given both of these vignettes, only twenty percent were able to transfer the lesson from the first story to the second (aim many of the low-intensity rays at the tumor from different directions and fire them all at once).

But when students were then reminded to think back to the general’s story, then a whopping seventy percent of students were able to solve the medical conundrum. After all, abstracting the two scenarios yields many similarities:

That’s great news! That shows that students, when guided, can transfer knowledge from examples. But it’s bad news because fifty percent of students wouldn’t have made the connection otherwise.

In the book Five Teaching and Learning Myths—Debunked by Adam M. Brown and Althea Need Kaminske, chapter two interrogates the efficacy of examples when teaching. Citing Gick and Holyoak, Brown and Kaminske suggest that the best way to leverage examples when teaching is to provide at least two different examples.

For instance, if another example had been given to the students that highlighted a city’s transportation system (building multiple roads that connect suburbs to the bustling downtown to balance the traffic load), students may have been more autonomous in their consideration of the tumor problem.

Having multiple examples helps students differentiate the surface details from the structural details. When students learn about the fortress solution, they may not initially recognize that load balancing is the solution since there are many other signals that can be confusing (for example, land mines, smaller troop numbers might be more stealth, defending a 360 degree attack is difficult). But adding an example from another domain, such as roads entering a city, help learners abstract the problem.

In fact, roughly fifty percent of students are able to transfer knowledge to a problem when given two examples (which is more than double the number of successful students when given only one example). And there was more improvement (around sixty percent) when students were provided with a principle or a diagram that helped explain the process (page 15).

Check out this passage from the book (page 21):

This basic rule can be applied even at the earliest stages of schooling. When explaining or describing a new concept use multiple examples. For example, if students are learning about fractions it is important to give examples of fractions in a variety of different contexts. Sharing a candy bar, pieces of a pie, proportion of marbles, making change from a dollar, etc. Use Different Examples: In the example above the multiple examples demonstrate slightly different aspects of fractions. Pies and candy bars are whole objects that can be broken into smaller pieces. Marbles are individual units that can be counted out as a portion over the whole set. Finally, money represents something in-between that has to be exchanged to divide up a larger unit into smaller units (i.e. a dollar can be divided into four by exchanging it for four quarters). By exposing students to variations of examples it helps them better understand the underlying concepts.

Brown and Kaminske also warn of seductive details – things that are interesting and will probably be remembered but distract from the actual problem. When I was in college, I had a math professor with spellbinding storytelling skills. I recall a story he told about two famous mathematicians who dueled because one of them published work he stole from the other one). I can tell you who won and what the fight was about – and this is 23 years later – but I can’t tell you how to do the LaPlace Transforms, which was the topic of the class.

I latched on to the sexy hook of a duel instead of learning how to do math. Seductive detail.

You should pick up this book – it is around seventy five pages and examines multitasking, examples, focus, testing, and learning styles. It’s a great read that provides actionable methods for teaching based on a lot of research in the field.