In my mathematics classes in CÉGEP, a lot of the content became more and more abstract and theoretical as I learnt more and more about calculus. As such, it became easy to lose perspective of what I was actually doing, since a lot of it was simply symbols. Fortunately, my teacher understood this and gave us plenty of examples to learn from.
I’ve been thinking about this topic in the context of first learning content. Here’s a question: should you start by giving students definitions, or examples?
Arguably, the former gives students a more complete view of the mathematics. After all, a definition is created precisely because people wanted to know when a certain concept applied or not. Therefore, a definition should be useful for students to figure out precisely what does and does not apply in the situation.
While that may be true, it’s probably not the way to introduce a new concept to students. I doubt I’m an outlier in saying that I always enjoyed seeing a graph or a picture of what we were doing in my mathematics class versus getting a rigid and complete definition. It’s not that the latter was bad, but that it didn’t give us any sort of intuition as to what we were doing.
For example, when I began to learn about derivatives and integrals in my calculus classes, my teacher didn’t immediately jump to showing the general definition of a derivative or integral. Instead, constant examples were provided, giving us a feeling for what these two mathematical concepts were. It wasn’t general by any stretch of the imagination. Instead, it was specific, and it gave us a foothold in the concept. Then, my teacher was able to step back and give us the general definitions.
In many disciplines, the opposite approach is what generally works. First, you focus on the general concepts, and then you drill down into the specifics for your goal. However, it’s easier to do the reverse in mathematics, because seeing an example (preferably, an easy one), allows a student to look at the example and think, I get the idea, how can we generalize it?
This is so crucial in the beginning of learning a mathematical concept, because definitions can be daunting. Often, they look like a bunch of symbols that just don’t make sense, whereas one can concretely get the idea of concepts like the directional derivative or the gradient vector by looking at examples or drawing graphs. Sure, they can also be understood by definitions, but it’s much more difficult to process. In general, we excel at absorbing information in a visual way, so it makes total sense that we would look at examples before generalizing.
The key idea here is that mathematics is about zooming out. We want to generalize information, formulas, and theorems as much as possible. It’s much more satisfying to have one equation that covers a whole gamut of possibilities instead of having one specific equation for each possibility. However, grasping these general equations aren’t immediately easy, since definitions are abstract. Therefore, the use of examples before broadening the picture allows students to understand what is happening for a specific concept, giving them a better idea of what the different components of a definition or theorem do.
If you just throw definitions and theorems at students, they will have a much more difficult time to grasp a concept than if you draw a simple graph or diagram.