> every working mathematician needs to understand at least basic set operations, simple identites like De Morgan's laws (I don't care if you know the name or not), as well as what cartesian products, relations and functions are.

This is stuff that you learn in the 5th or 6th grade in school, and is about as far removed from what mathematicians call "set theory" as basic arithmetic operations are from college math courses for soon-to-be mathematicians.

You may have learned this in 5th or 6th grade, but I certainly didn't.

The point is that any working mathematician is comfortable manipulating sets in algebraic expressions and that's not something you expect from your average high schooler.

I could have added that mathematicians need to know at least about different cardinalities, but I guess strictly speaking you could be working in discrete maths and not care about any of this.

that is a little bit of hyperbole. No school in 5th or 6th grade is giving formal definitions of relations and functions. There is a reason nearly all undergrad math books in analysis, topology, algebra, all devote an entire first chapter to it.

> No school in 5th or 6th grade is giving formal definitions of relations and functions.

I remember that my math teacher pretty surely did.

> There is a reason nearly all undergrad math books in analysis, topology, algebra, all devote an entire first chapter to it.

Indeed there exist multiple good reasons:

- recapitulation

- setting up the notation

- clarifying how the textbook defines the relevant mathematical objects, because the definitions of some concepts might differ depending on the textbook

- making clear what existing standard the textbook expects from the learner

- ...

how do you motivate formal functions and relations to 5th graders, and more importantly why? Relations are important because of the natural partitions of a set they create and the development of group theory. Functions are useful in calculus, but not really the algebraic properties, those are glossed over, i.e kids learning calculus are usually not learning the formalization of functions. That isn't important until analysis or abstract algebra, hence why its included in the textbooks

> how do you motivate formal functions and relations to 5th graders, and more importantly why?

The word "you" in English has two meanings:

1. how would I motive this to 5th graders ("you" as "tu/vous" in French or "du/Sie" in German)?

2. how is this topic motivated in school to 5th graders ("you" as "on" in French or "man" in German)?

For 2: Well, it isn't. The pupils have to accept that in future, they will hopefully get why it is useful. Until then, better learn the material so that you won't fail on the tests.

For 1: If the prophet does not come to the mountain, the mountain must come to the prophet. If you need group theory to motivate functions (as you implicate in your answer), then teach group theory to 5th graders, so that the pupils get the motivation that they desire. If you additionally need to motivate group theory: well, I do know some quite interesting applications of group theory. :-)

Just to make it clear: I do have quite some experience in teaching mathematics, but to highly gifted students.

> Functions are useful in calculus, but not really the algebraic properties, those are glossed over, i.e kids learning calculus are usually not learning the formalization of functions. That isn't important until analysis or abstract algebra, hence why its included in the textbooks

In Germany, there is no distinction made between calculus and analysis. At the university, this subject is taught from beginning on in the abstract way. In school, what you call "calculus" is often taught in a more "hand-waving" way by bad math teachers. Good teachers rather attempt to teach calculus/analysis in the abstract way in school.

fair enough, I was only speaking for the US, which has a slower pace and lack of motivation throughout k-12 + calc series. I still have no idea why they don't teach basic algebra with matrices, so it doesn't seem like a giant bad of tricks, or why linear algebra is so separated from multi-dim calculus, it really makes it more difficult with busy work and you never really comprehend anything.

It would be dangerous to teach anyone in ML what a function was, since they might realise, the premise of the entire activity is false.

(ie., that in the vast majority of cases there are no empirical functions to model, no f: Pixel -> Animal)

So better continue the current practice where few could distinguish a relation and a function; and fewer still are aware that they arent approximating empirical functions.