An SL Mathematics teacher marking the exam papers asked me: is it ok to draw the 3D axes in any orientation? To understand this question, this is how the textbooks we use always draw the 3D axes:

The textbook even describes this as "the X-axis is considered to come directly out of the page". So this teacher wanted to know whether to deduct marks for students who drew it like this:

I thought there's no problem at all because, well, it depends on the perspective, right? If you turn your head a bit (in 3D space of course), surely they're the same? But this teacher insisted on getting a consensus, made me ask another teacher who taught IB before, and after both of us assured her there is no problem, she went on to ask another HL teacher (who told her the same thing).

Well, I'm not criticising this particular teacher, who happens to be a good friend and a nice person. It's just ... that mentality that is so unhealthy in education. She kept insisting that in O level there're certain conventions we have to follow and she wants to know whether this is an issue in IB. Well, I think even conventions must have a good reason, unless it's semantics (in which case we should call a rose a rose, and not something else, and it'd be silly to ask why we call it a rose). Just because a textbook prefers one orientation does not make it a convention. And as mathematicians, we ought to be able to judge whether it really is a reasonable convention!

Which reminds me of this incident while I was teaching English during my practicum: the students told me they were told NEVER to write a ghost story or a dream for composition. I never figured out whether they misconstrued the meaning of their teachers, or this is really what they were told. I think we should NEVER tell students NEVER to do something, but rather explain to them how doing these things may produce undesirable effects.

For the record, something like this may be considered wrong:

"Wrong" axes

This is wrong if you adopt the cross-product convention in vectors. For those who may understand: we should follow the right-hand rule, thus in this case if we keep the x and y axes orientation, then the z axis orientation should point "inward" rather than "outward". Even then, this is not an issue that can be compared to calling a rose a rose. It's more like arguing whether collaborative learning is different from cooperative learning. According to some scholars, yes, according to others, no. And it's just a matter of how you define the 2 words. This simply illustrates the fact: the real world is full of ambiguities. We need to clarify if there're ambiguities but we also need to learn how to live with it!