Why Are Infinite Concepts Misunderstood in Math Education?

[loom91]

Who said anything about justifying limits as 'approximate or subjective'? The fact that derivatives are used to linearize and approximate is what I was talking about and has nothing to do with what you think I said. Dont' get me wrong, I wish that maths were taught properly, but I don't think the problems you perceive are to do with poor understanding of the concepts by teachers. The course content is not decided by the teachers, or the mathematicians, but by the engineers who teach the output of the course. I think you're ascribing too much mathematical sophistication to the average student as well.

Further, I suggest you look up the space of dual numbers before saying much more.

This, also, makes little sense:

"as far as I know e^2 is a uniquely defined quantity that, being the limit of a strictly increasing positive sequence, is necessarily non-zero"

What limit of what strictly increasing sequence? e is just a symbol, and the ring of dual numbers is just the the ring k[e]/e^2. It is how algebraists do the algebraic version of differential geometry without taking limits which are usually meaningless in algebraic contexts.

I don't know what interpretation of e^x you are talking about and what meaning can be assigned to the statement e^2 = 0, but I don't think dual numbers have any relevance here. In the context of calculus, which we have been discussing here, e^2 is defined as the limit of a sequence of partial sums, and this limit is not 0.

I also don't think poor teaching is due to teachers not undersanding, it is due to them not teaching. I'm sure most teachers have a good understanding of what limits are, but as long as they are teaching wrong concepts to the students their internal knowledge is of little relevance.

I don't know why you are bringing engineers into this, I repeat that I'm not talking about engineering courses. It also has little to do with course content, I'm talking about teaching methods. As our chemistry teacher used to say, students lives are being destroyed because the lords of education are eternally afraid that too much stress will be placed upon the fragile minds of the students. The whole process is like trying to pull back the steering more to generat lift when the wing has already stalled: the situation goes from bad to worse.