by Doris Carey, Faculty Member, Faculty of Academic and Career Preparation, VIU

When introducing a new concept, instructors tend to provide a concise definition of the term and follow up with a few examples of the concept. For example, if I’m introducing a lesson on polynomials, I make a list of algebraic expressions with three, four or five terms. I also explain that monomials and binomials are polynomials and add a few of those to the list. We all agree the concept is quite easy.

Then comes the hard part of understanding a concept deeply: distinguishing examples of the concept from non-examples. Students have no reason to suspect that 5/x isn’t a polynomial. They don’t necessarily go back to the definition of the concept. Because it looks like a monomial, they think that 5/x could fit in with the list of examples I’ve given and can’t see any reason to exclude what appears to be a good candidate for membership in the class of polynomials.

What went wrong?

Studies of how very small children learn new concepts have given us a great deal of insight about how we all learn new concepts. The studies reveal that toddlers and even babies who have not yet developed enough vocabulary to communicate their understanding are able to pick and choose among the artifacts in their environment and classify items.  The way they gather the information they need to understand a concept is of primary importance.

For example, toddlers hear adults talking about cars all the time. “Let’s go for a ride in the car,” or “I’m going to go wash the car now,” or “Sally bought a new car.” When a car goes by, adults point to the car and the child might say, “car,” in imitation of the adult. As time goes by, the child might point to any vehicle and say, “car,” but an adult might say, “No, that’s a truck” or, “That’s a bus.” They continue to test their understanding of the word by adding examples of cars, and are able to eliminate things that turn out to be buses and trucks and vans even though all of those have four wheels, windows, passengers and shiny coats, just like cars do. In finding negative examples, children distinguish the salient characteristics of the concept from characteristics that are non-essential. This is how we naturally refine our understanding of new concepts.

To return to the study of polynomials now, we can see that students were given multiple examples of polynomials so they could observe the characteristics that give specific algebraic expressions membership in the class, but they were unable to correctly exclude 5/x because they had no way to determine what salient characteristics were missing or to find out what characteristics the example might possess that eliminates it from the class of polynomials.

The introduction of a new concept does require many positive exemplars, but for deep understanding, the lesson must include a number of well-chosen negative examples, along with an opportunity to identify one or more attributes that lead to the exclusion from the class. When I studied history in high school, my teacher described “democracy” by listing a few characteristics of systems that had democratic governments. I didn’t fully understand democracy (let alone appreciate it) until I was able to compare it with a plutocracy, a dictatorship, a monarchy or other forms of government.

As a researcher, I have observed classroom environments in which students were given rich sets of data that they could use to identify the essential attributes of a concept. I’ve watched the actions of students who were used to examining lists of positive and negative examples in the formation of a concept. When these same students sat in traditional classrooms where concepts were introduced strictly by definition and limited positive exemplars, they were able to aggressively seek out the negative examples they needed to round out their learning. “Miss G., is 21 an example of a prime number?” Forcing a negative example was a way to get more information from the teacher: “No, 21 isn’t prime because 3 and 7 are both factors of 21.”

In our data rich world, students can probably figure out how to Google, “What’s an example of a country that isn’t a democracy?”