The « Heard-of-it Effect » in Learning

This entry was first posted on my old blog page, on September 25 2014.

Today was the premiere of the second episode of my documentary on (French) undergrad maths. I’ve had plenty of feedbacks already, with fundamental questions about the way I popularize maths.

Roughly, while the narrative of the documentary is quite understandable, I’ve decided to include high-level interviews. For instance, in the first episode available online, one interview sequence is about abstract algebras. In particular, interviewees show great excitement regarding so-called groups, rings and fields, which are not defined (groups are defined shortly after). Thus, if you haven’t learned about these mathematical structures yet, you will be lost in this interview sequence. I know it. I planned it.

Granted, viewers may reject the whole of maths, because they feel that they will never be able to understand this section. Yet, that is the opposite of the asserted aim of the video, which is rather about making maths attractive. I know that. And that deeply bothers me. So why did I plan to have the audience confused? There are several reasons for this.

First, I wanted interviewees to get excited about maths, and, as weird as it might sound to you right now, such structures easily got (most of) them excited. And I think it’s pretty cool to hear mathematicians getting excited. Second, mathematics is hard. And mathematicians spend the overwhelming majority of their time trying to make sense of what they don’t understand. To enjoy mathematics definitely requires being comfortable with not understanding. Or at least, that’s what I think. Third, anyone who does know about these structures will greatly enjoy these sections.

But more importantly, I want to plant seeds of what I call the « Einstein effect ». Why did you read this post in the first place? I bet it’s because I mentioned « Einstein » in the title… and you’ve heard somewhere that Einstein was a big deal. In fact, back in undergrads, I distinctly remember the relativity theory course being full, as opposed to the Galois theory one. Why? Is it because relativity theory is more useful or more beautiful? I don’t think so. Relativity theory is useless for all engineers, and the Galois theory is arguably the most beautiful mathematical theory ever. So, what makes relativity theory more attractive than Galois theory?

I think it’s because people have been more exposed to physicists’ excitements about Einstein’s work than to mathematicians’ trepidations regarding Galois’. And this is essential in the learning process! Think about the most boring math course you’ve had. If, at some point, someone told you that what you were learning was a key concept for understanding Einstein’s relativity, this would probably awake you and give you a small boost. This boost is what I call the « Einstein effect ». And it may be just what you need to be willing to do what’s necessary to grasp the concept (that is, spending hours pondering).

Similarly, I think that if a student has been exposed to the importance of groups, rings and fields or Galois theory, he might have this necessary additional boost to pay attention to the professor as the professor defines these concepts… That’s why, while talking about difficult concepts in a popularization video may be negative in the short run, I believe it to be very useful in the longer run. Besides, it’s also why I think we shouldn’t only teach numbers and basic geometry to kids. We should also tell them about infinite series (like 1+2+4+8+16+…=-1), probabilities (like the Monty Hall problem) and topology (like Pacman living in a torus), even if they don’t get it all!

What do you think?

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