In the last lesson, we discussed analogies. Analogies involve two seemingly different situations that share an underlying deep structure. This lesson is about a close cousin to analogies: contrasting cases. Contrasting cases also involve variation among two examples, but in this case, the examples share many features in common. With contrasting cases, it is the distinctions between the two that are important. Experts have the ability to make fine distinctions, and contrasting cases is great for developing that skill.
Discovering Fine Distinctions
Contrasting cases change how we “see.” If you want a vivid example of this, try to draw a penny (or comparable coin). What (and where) is the text on the penny? Where is the bust and which direction is it facing? Now take a look at an actual penny. If you’re like most people, you haven’t paid much attention to the penny even though you’ve seen them your whole life. Why? The reason is that you have only ever had to distinguish pennies from other coins: nickels, dimes, and quarters. The penny is pretty different. It’s copper and the others are all silver. That’s about all you need to remember about it to use it effectively. As a result, there was no need to see all the fine details on the penny, unless you are a coin collector.
Without contrasts, it’s difficult to see which parts of an example are important. You see one car and it’s just a car. It’s got all of the car-like things: windshields and side view mirrors and headlights. But when you see two or three cars together, you can see what’s different about the cars (and what they have in common): One has a spoiler—does that make it go faster? One is lower to the ground—does that make it more stable?
There’s something quite powerful about seeing the examples together, at the same time. Another example is below.
On the left are three different correct ways of simplifying a math expression. On the right is one incorrect way. For a student learning to simplify math expressions, seeing these examples together can be a powerful way of both seeing what’s common about the correct solutions (they all perform the multiplication step between the 2 and 4 before adding them to the rest of the values) and what’s different about the incorrect solution (4 is added to another value before multiplying by 2).
Both analogies and contrasting cases emphasize that “two is better than one.” A single example is just not enough for us to learn what we need to know.
How Can These Ideas Help?
The mere act of putting closely related examples together on a page is quite helpful. If you’re learning mathematical formulas, put multiple similar-looking problems together on the same page, and try to figure out why one formula is used in some cases and another in others. When learning a Chinese character, compare it to similar looking characters—that will get you to notice the difference (rather than re-reading it or writing it over and over again).
It’s not enough to see positive instances: “This is a dog.” You need to see negative instances as well: “This is NOT a dog—it’s a cat.” It’s best to drill down to finer and finer distinctions as you learn. We start recognizing the distinction between dogs and cats as toddlers. The next logical step, if we wanted to become a dog species expert, would be to make broad comparisons—a poodle to a German shepherd—then finer distinctions—a greyhound to an Italian greyhound. This iterative approach is more effective than trying to look at groups of cases that differ in many different ways.
Learning Chinese characters offer a perfect case of contrasting cases. For example, four characters: “friend,” “right,” “left,” and one meaning (roughly) “to go out” all look very similar. By looking at them together, I can observe the important differences. Is there a way for you to use contrasting cases to observe fine distinctions between things that look similar but are different?
We move on from prior knowledge, analogies, and contrasting cases to a new topic next lesson: practicing in the right way.
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