As you learn more ideas in mathematics, it’s easy to start feeling like certain ideas are “below” you. This often comes in the form of saying that ideas are “trivial”, as if they shouldn’t take up any of your time. This can be exacerbated further in mathematics by the idea of rigour. Once we learn that not all proofs are equal, it can be tempting to say, “Okay, I get this proof, but that’s not the whole story. You’re not being fully rigorous here.” We can then get caught in the cycle of thinking of our work as more important than “basic” facts.
One place where I think this plays out is in the use of animations, diagrams, and heuristics to teach a topic. Here, I’m using the term “heuristics” as an umbrella term that encapsulates anything that wouldn’t be called a strict rigourous proof. This might mean using words instead of precise mathematical statements, or diagrams like I mentioned above. The point is that we substitute some of the abstractness in order to get at the core idea.
The problem I have with people who dislike this sort of learning by analogy is that I don’t think heuristics are ever the end goal. Sure, you might use a nice diagram to get an idea of what is happening within the mathematics, but it’s not the whole story. If you ask someone who specializes in making animations or diagrams that illustrate certain mathematical concepts, I doubt they would tell you to stop once you understand their explanation. Instead, they will tell you to dig deeper to understand all of the nuance that’s available. In fact, I suspect that they would insist you do this, because that’s where a deep understanding comes from. It’s not enough to just go at the surface level. To really learn, you have to go deeper.
Think about it from the perspective of the creator. They are attempting to take a complex idea and distill it down into a heuristic that is easier to grasp. It takes a ton of work to simplify ideas. However, when they are done this process, they still understand the inner details that may be hidden. That’s because they have done the difficult work of understanding the whole structure before reshaping it for others. As such, they know both the heuristic and the actual details. This is a powerful place to be.
For the audience though, they begin with the heuristic. That’s not a problem, but I think we should be more upfront with how we encourage people to dive deeper into a topic. Sure, the heuristics can give a person some taste for the idea, but the real fun starts when you go further and see how all the details play out. That might not be evident from the first exposure, but it’s something you can absolutely improve on.
I don’t think there’s anything wrong with using heuristics. Sure, they may not be the full story, but in order to prepare yourself for the full story, you need to learn the basics first. This is where heuristics really shine. They can give you an overview that lets you decide how much further you want to go. However, I think we need to also acknowledge a second function: heuristics can get you engaged with an initial idea more than the pure mathematics probably will.
Heck, we start learning by using heuristics. When we first learn mathematics, we aren’t talking about epsilons and deltas. We don’t begin our mathematics education with, “Let S be a set.” We start with familiar ideas, we build analogies, and we slowly learn how those initial ideas can be expanded and made more rigorous using mathematics.
There’s a place for both rigour and heuristics in mathematics education. Roughly speaking, I would say that you end up getting a “heuristics sandwich”. When first learning, you use heuristics. Then, you get sophisticated enough to understand the full details. Finally, once you’re really comfortable, you fall back to the heuristics because they are easier to reason with. There’s nothing bad with either of them. Rather, it’s useful to have a bit of both.