5 ms·
Humans don’t learn from abstract definitions: I agree very very much with that. The definition comes later. When people learn math concepts, the definition is
by anyfoo 5y ago
Humans don’t learn from abstract definitions: I agree very very much with that.
The definition comes later. When people learn math concepts, the definition is in fact often of surprisingly little value. They need to see examples, make examples, play around with instances and applications of the “thing”, to gain intuition and finally come back to the definition.
The abstract definition is also often (usually? always?) something that the person who came up with it also only came up with after finding the abstraction of otherwise concrete things they were working with.
Try understanding Fourier transforms, or worse, Laplace, from their definition alone. Unless you have an idea of what’s happening already, it’s extremely hard. Work with the concept for a while, and the definition appears laughably simple!
Once you did that, look up the definition of a Hilbert transform. What is that?! But in practice, it’s just “multiply all complex frequencies by j” or, much more intuitively: “rotate all phasors by 90°… i.e. put them on their side”. Unlike with the previous example, you might happily work with this one and use it a lot, and still be mystified by its definition.
And, yes, I only finally understood and later fully embraced Monads by “just using them” for a while. Now the concept is clear and they are everywhere. Use them even more, and you run into situations where Monads are not enough in practice. And now you’ve made the first step towards Arrows!
- bonoboTP 5y agoOn the other hand, I hate it when people take this mentality to the extreme and withhold the mathematical definition and insist on only using examples or "simplified" definitions. It's similar with language learning. Sometimes I just want to see the grammar rule (there often is one, at least to a very good approximation), but the teacher has this didactic philosophy that I wouldn't be able to "handle the truth" or something. If there is an adequate definition provide it. It's similar with computer science papers as well. On the one hand, a free-text description of an algorithm or a simplified pseudocode is helpful to get the gist, but very often I also need to see the actual unambiguous code to see exactly how it is and to check if I understood it right because no description covers all the corner cases. Both are important, practical examples but also the abstract definition (if one exists to a good approximation).
- anyfoo 5y agoThat's also bad, yeah. We came to the abstract definition because it has value after all. (I think) OP and me just stated that it's not necessarily useful or sufficient to try to understand starting from the definition, but I do think there is still value in even just stating the definition already, for reference. Luckily I can't remember encountering that didactic approach in math or language learning, but I'm not surprised at all if it exists and is somewhat widespread.