One framework of understanding and building towards proof that we teach to mathematics teachers at primary school level is (and translated to English on fly):
- Naive empiricism: The pupil tries an example, or maybe several, and checks if the claim holds for them.
- Crucial experiment: The pupil tries an example chose with care such that if something should be wrong with a claim, this example would show it. The cruciality is not formally proven, however.
- Generic example: Here one takes a representative example and uses it to show an argument that works for any example, but this generality is not formalized (by, for example, using variables).
- General argument: An idea of proof, a mathematician would say.
- Formal argument: A mathematical proof.
There is some overlap between these, and some of these are also proof strategies when formalized.
I think one to three are directly from
Balacheff, N. (1988): Aspects of proof in pupils’ practice of school mathematics,.
En Pimm, D. (ed.), Mathematics, teachers and children (Hodder & Stoughton: Londres),
His fourth category might be a bit different than the ones I use here.
Looking at the video, the presenter demonstrates the claim with a single example and invites (claims) it to hold for others, too. So we are at least at the level of naive empiricism.
We might argue that there is nothing special about these particular cylinders (they seem to be of different ratios of height to radius) and it is also unlikely that something whether this ratio is a rational number or not, or some other occult property, would have any effect. So maybe this is a generic example? But the video does not argue this; I am over-interpreting here. But not much.
We did not see this done with a cylinder that has very extreme form, like ratio of height to radius being very small or very large. Neither did we see the exercise done with a great deal of accuracy. Had we seen such, maybe we could say that this amounts to a generic example: the claim would have been demonstrated with a usual, an extremely sharp and extremely dull object, and thus it would have seen likely that it also holds for other shapes. Alas.
Se we certainly saw a sloppy example, and were implied to that this same should hold generally, so we are at least at naive empiricism and maybe, maybe, seeing a generic example.