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Thought a little background is required.

The place I teach students learn Calculus in school, not the theory but only the computational aspect.

I am trying to teach Calculus to first year Engineering students using Apostol but there is huge resistance from my own Department. The arguments range from Engineers don't need this, they won't understand theory and its useless for them, they should focus more on concepts and visualization (geometric intuition). I am feeling a bit defenseless and cornered as I never been part of discussions where people discussed pedagogy.

All I know, I spent 15 odd years in an Engineering Institute and 95% percent of my friends are Engineers and they mostly know/did their maths courses rather well; which was a proper course, as in they did proofs, knew about the real number system (lub axiom) , Sequences (Bolzano Weierstrass) , Continuity, Roll's Theorem and MVT, Taylor's Expansion all, Fundamental Theorem of Calculus and they did all these in their first semester along with proofs.

Here I am facing a scenario where Cauchy Sequence is not to be touched on as if it is going to make students head explode. Taylor's Expansion is not to be talked about, forget Taylor's Series I am not allowed to teach Series. The problem is the resistance comes from my own head/department and not the Engineering Department.

I am totally confused and do not know how can I argue my case. Apparently all these is part of Real Analysis and not Calculus and also Proofs are not required, how does one argue against this ?

I am not complaining at all, I am asking for advise regarding how to present my case in a compelling and convincing way.

Regards Vb

Added in Response to comment by Douglas Zare

those with different preparation would probably go through a preparatory course I suppose. What I mean to what extent do the Engineering students need to know Calculus to do well in their Career. What is the basic minimum. Where they begin is a different issue, what's the least they must & should know ?

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migrated from mathoverflow.net Sep 5 '14 at 12:31

This question came from our site for professional mathematicians.

  • $\begingroup$ Please convert this to Community Wiki $\endgroup$ – Vagabond Sep 5 '14 at 10:18
  • $\begingroup$ I think that the answer to this is hugely going to depend on where you are situated. Are you in the United States? Canada? Germany? China? Russia? $\endgroup$ – Simon Rose Sep 5 '14 at 10:22
  • $\begingroup$ Hi Simon, Its India where I teach. $\endgroup$ – Vagabond Sep 5 '14 at 10:25
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    $\begingroup$ It shouldn't just depend on the country. Within the US, it would depend on the backgrounds of the students, and what the students need to get out of the course. Those vary greatly from school to school. $\endgroup$ – Douglas Zare Sep 5 '14 at 10:25
  • $\begingroup$ Hi Douglas, I agree, but then those with different preparation would probably go through a preparatory course I suppose. What I mean to what extent do the Engineering students need to know Calculus to do well in their Career. What is the basic minimum. Where they begin is a different issue, what's the least they must & should know ? $\endgroup$ – Vagabond Sep 5 '14 at 10:36
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I strongly agree with Douglas Zare that it is highly dependent on what incoming students are prepared for in terms of their mathematical background beforehand. As far what is the minimum calculus knowledge that engineers need, this too is dependent on what field of engineering they go into. I know several engineers that use no more than basic trigonometry in what they do and do not need to know much (if any) calculus, though that will certainly not be true for all engineers.

In the US (where I teach), many schools make a distinction between a calculus course and a real analysis course. A calculus course is essentially a problem-solving course, in which students are introduced to basic facts related to the operations of differentiation and integration, with a few supplementary facts about underlying notions such as the real number line and limits, and then expected to be able to solve a wide range of problems that, in my view at least, often range from essential calculations that everyone should be able to do to esoteric trivia that has remained in the curriculum due to inertia. On the other hand, a real analysis course should be a careful introduction to the real number line and functions of a real variable, based on rigorous proofs.

Typically, engineering students are required to take the sequence of calculus courses, but not required to take an analysis course. We generally push the stronger students to continue with the analysis sequence, and their "reward" for doing so is the opportunity to add a second major in mathematics, which has a non-trivial amount of employment value.

This system suggests that analysis is not required in a bare minimum of what a prospective engineer needs to know, but that it is a valuable asset that some engineering firms will value (either directly for the subject knowledge obtained or indirectly as a signaling mechanism to indicate an ability for abstract thought).

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For one thing, you can argue that students need to know separation of variables http://en.m.wikipedia.org/wiki/Separation_of_variables and therefore need to know Series.

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    $\begingroup$ The link between separation of variables and series is far from obvious. $\endgroup$ – Rory Daulton Sep 5 '14 at 13:16

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