I do not think this is a very good argument. The„smartass kid“ is of course only making the life of the teacher even more exhausting while not providing insight on their own with their question. However, I still think they have a very important point which is missed here: Our way of teaching mathematics (beyond arithmetic and elementary geometry) in school provides next to no value to the vast majority of people. Of course a society where mathematics exists is preferable to one where it doesn’t, but whether or not there are people who are creating value for society using mathematics has nothing to do with school.

People who won’t use mathematics later in life do not even get to know what mathematics even is or how it _might_ and _does_ enrich their lives, while those who will have to figure that out on their own and can only then start to actually learn the mathematics behind the math class (on their own or in university). (I for example found mathematics classes extremely boring and later on even infuriating for having to see how the beautiful algebra and analysis was taught to other students.)

Of course, I firmly believe it is _possible_ (and important!) to make mathematics classes a reality where the teacher is in fact correct, where the teaching of mathematics actually provides insight for everyone, regardless of application, while also making aware of the power it has for applications. In such a world, the question raised by our kid here is then a way for the teacher to know that this goal was not achieved for all students in class yet. Ideally, then, the teacher would not react in a condescending way to the question, but instead embrace that there are applications (of course, it’s the area of a triangle!) but also that there need not be a direct application situation for the lesson, if listened to, to be valuable.